Theorem recnprss 25820

Description: Both ℝ and ℂ are subsets of ℂ. (Contributed by Mario Carneiro, 10-Feb-2015.)

Assertion

Ref	Expression
recnprss	⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)

Proof of Theorem recnprss

Step	Hyp	Ref	Expression
1		elpri 4646	. 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ))
2		ax-resscn 11187	. . . 4 ⊢ ℝ ⊆ ℂ
3		sseq1 4003	. . . 4 ⊢ (𝑆 = ℝ → (𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝑆 = ℝ → 𝑆 ⊆ ℂ)
5		eqimss 4036	. . 3 ⊢ (𝑆 = ℂ → 𝑆 ⊆ ℂ)
6	4, 5	jaoi 856	. 2 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝑆 ⊆ ℂ)
7	1, 6	syl 17	1 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1534   ∈ wcel 2099   ⊆ wss 3944  {cpr 4626  ℂcc 11128  ℝcr 11129

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-ext 2698  ax-resscn 11187

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-un 3949  df-in 3951  df-ss 3961  df-sn 4625  df-pr 4627

This theorem is referenced by:  dvres3  25829  dvres3a  25830  dvcnp  25835  dvnff  25840  dvnadd  25846  dvnres  25848  cpnord  25852  cpncn  25853  cpnres  25854  dvadd  25858  dvmul  25859  dvaddf  25860  dvmulf  25861  dvcmul  25862  dvcmulf  25863  dvco  25866  dvcof  25867  dvmptid  25876  dvmptc  25877  dvmptres2  25881  dvmptcmul  25883  dvmptfsum  25894  dvcnvlem  25895  dvcnv  25896  dvlip2  25915  taylfvallem1  26278  tayl0  26283  taylply2  26289  taylply2OLD  26290  taylply  26291  dvtaylp  26292  dvntaylp  26293  taylthlem1  26295  ulmdvlem1  26323  ulmdvlem3  26325  ulmdv  26326  dvsconst  43690  dvsid  43691  dvsef  43692  dvconstbi  43694  expgrowth  43695  dvdmsscn  45247  dvnmptdivc  45249  dvnmptconst  45252  dvnxpaek  45253  dvnmul  45254  dvnprodlem3  45259