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Mirrors > Home > MPE Home > Th. List > nelneq | Structured version Visualization version GIF version |
Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.) |
Ref | Expression |
---|---|
nelneq | ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2902 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | 1 | biimpcd 251 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶)) |
3 | 2 | con3dimp 411 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2816 df-clel 2895 |
This theorem is referenced by: nelne2 3117 onfununi 7980 suc11reg 9084 cantnfp1lem3 9145 oemapvali 9149 mreexmrid 16916 supxrnemnf 30495 onint1 33799 bj-fvmptunsn2 34542 maxidln0 35325 rencldnfilem 39424 climlimsupcex 42057 icccncfext 42177 |
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