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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrgtned | Structured version Visualization version GIF version |
Description: 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrgtned.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrgtned.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrgtned.3 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
xrgtned | ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrgtned.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xrgtned.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | xrgtned.3 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
4 | xrltne 13168 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | 1 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 ℝ*cxr 11271 < clt 11272 |
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-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
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-nel 3042 df-ral 3057 df-rex 3066 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-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 |
This theorem is referenced by: xrge0nemnfd 44686 xrltned 44711 infxr 44721 pimxrneun 44843 liminflimsupxrre 45177 ioorrnopnxrlem 45666 gsumge0cl 45731 sge0pr 45754 sge0rpcpnf 45781 sge0isum 45787 |
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