Theorem rankeq1o 35703

Description: The only set with rank 1_o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)

Assertion

Ref	Expression
rankeq1o	⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅})

Proof of Theorem rankeq1o

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8502	. . . . . . 7 ⊢ 1o ≠ ∅
2		neeq1 2998	. . . . . . 7 ⊢ ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
3	1, 2	mpbiri 258	. . . . . 6 ⊢ ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
4	3	neneqd 2940	. . . . 5 ⊢ ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5		fvprc 6883	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
6	4, 5	nsyl2 141	. . . 4 ⊢ ((rank‘𝐴) = 1o → 𝐴 ∈ V)
7		fveqeq2 6900	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8		eqeq1 2731	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 1o ↔ 𝐴 = 1o))
9	7, 8	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → (((rank‘𝑥) = 1o → 𝑥 = 1o) ↔ ((rank‘𝐴) = 1o → 𝐴 = 1o)))
10		neeq1 2998	. . . . . . . 8 ⊢ ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
11	1, 10	mpbiri 258	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12		vex 3473	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	rankeq0 9876	. . . . . . . 8 ⊢ (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
14	13	necon3bii 2988	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
15	11, 14	sylibr 233	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → 𝑥 ≠ ∅)
16	12	rankval 9831	. . . . . . . 8 ⊢ (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
17	16	eqeq1i 2732	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18		ssrab2 4073	. . . . . . . . . . 11 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19		elirr 9612	. . . . . . . . . . . . . 14 ⊢ ¬ 1o ∈ 1o
20		1oex 8490	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
21		id 22	. . . . . . . . . . . . . . 15 ⊢ (V = 1o → V = 1o)
22	20, 21	eleqtrid 2834	. . . . . . . . . . . . . 14 ⊢ (V = 1o → 1o ∈ 1o)
23	19, 22	mto 196	. . . . . . . . . . . . 13 ⊢ ¬ V = 1o
24		inteq 4947	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∩ ∅)
25		int0 4960	. . . . . . . . . . . . . . 15 ⊢ ∩ ∅ = V
26	24, 25	eqtrdi 2783	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
27	26	eqeq1d 2729	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
28	23, 27	mtbiri 327	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
29	28	necon2ai 2965	. . . . . . . . . . 11 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30		onint 7787	. . . . . . . . . . 11 ⊢ (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
31	18, 29, 30	sylancr 586	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32		eleq1 2816	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}))
33	31, 32	mpbid 231	. . . . . . . . 9 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34		suceq 6429	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1o → suc 𝑦 = suc 1o)
35	34	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36		df-1o 8480	. . . . . . . . . . . . . . . . 17 ⊢ 1o = suc ∅
37	36	fveq2i 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘1o) = (𝑅1‘suc ∅)
38		0elon 6417	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ On
39		r1suc 9785	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41		r10 9783	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1‘∅) = ∅
42	41	pweqi 4614	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 (𝑅1‘∅) = 𝒫 ∅
43	37, 40, 42	3eqtri 2759	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘1o) = 𝒫 ∅
44	43	pweqi 4614	. . . . . . . . . . . . . 14 ⊢ 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45		pw0 4811	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
46	45	pweqi 4614	. . . . . . . . . . . . . 14 ⊢ 𝒫 𝒫 ∅ = 𝒫 {∅}
47		pwpw0 4812	. . . . . . . . . . . . . 14 ⊢ 𝒫 {∅} = {∅, {∅}}
48	44, 46, 47	3eqtrri 2760	. . . . . . . . . . . . 13 ⊢ {∅, {∅}} = 𝒫 (𝑅1‘1o)
49		1on 8492	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
50		r1suc 9785	. . . . . . . . . . . . . 14 ⊢ (1o ∈ On → (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o))
51	49, 50	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o)
52	48, 51	eqtr4i 2758	. . . . . . . . . . . 12 ⊢ {∅, {∅}} = (𝑅1‘suc 1o)
53	35, 52	eqtr4di 2785	. . . . . . . . . . 11 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
54	53	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
55	54	elrab 3680	. . . . . . . . 9 ⊢ (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
56	33, 55	sylib 217	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
57	12	elpr 4647	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58		df-ne 2936	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59		orel1 887	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
60	58, 59	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61		df1o2 8487	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
62		eqeq2 2739	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
63	61, 62	mpbiri 258	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → 1o = 𝑥)
64	63	eqcomd 2733	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → 𝑥 = 1o)
65	60, 64	syl6com 37	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
66	57, 65	sylbi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
67	66	adantl 481	. . . . . . . 8 ⊢ ((1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
68	56, 67	syl 17	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
69	17, 68	sylbi 216	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
70	15, 69	mpd 15	. . . . 5 ⊢ ((rank‘𝑥) = 1o → 𝑥 = 1o)
71	9, 70	vtoclg 3538	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝐴) = 1o → 𝐴 = 1o))
72	6, 71	mpcom 38	. . 3 ⊢ ((rank‘𝐴) = 1o → 𝐴 = 1o)
73		fveq2 6891	. . . 4 ⊢ (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74		r111 9790	. . . . . . 7 ⊢ 𝑅1:On–1-1→V
75		f1dm 6791	. . . . . . 7 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On)
76	74, 75	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
77	49, 76	eleqtrri 2827	. . . . 5 ⊢ 1o ∈ dom 𝑅1
78		rankonid 9844	. . . . 5 ⊢ (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
79	77, 78	mpbi 229	. . . 4 ⊢ (rank‘1o) = 1o
80	73, 79	eqtrdi 2783	. . 3 ⊢ (𝐴 = 1o → (rank‘𝐴) = 1o)
81	72, 80	impbii 208	. 2 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = 1o)
82	61	eqeq2i 2740	. 2 ⊢ (𝐴 = 1o ↔ 𝐴 = {∅})
83	81, 82	bitri 275	1 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  {crab 3427  Vcvv 3469   ⊆ wss 3944  ∅c0 4318  𝒫 cpw 4598  {csn 4624  {cpr 4626  ∩ cint 4944  dom cdm 5672  Oncon0 6363  suc csuc 6365  –1-1→wf1 6539  ‘cfv 6542  1_oc1o 8473  𝑅₁cr1 9777  rankcrnk 9778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607  ax-inf2 9656

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-r1 9779  df-rank 9780

This theorem is referenced by: (None)