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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 2829 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
| 2 | 1 | biimpcd 249 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶)) | 
| 3 | 2 | con3dimp 408 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 | 
| This theorem is referenced by: nelne2 3040 onfununi 8381 suc11reg 9659 cantnfp1lem3 9720 oemapvali 9724 mreexmrid 17686 supxrnemnf 32772 elrgspnlem4 33249 elrspunsn 33457 onint1 36450 bj-fvmptunsn2 37259 maxidln0 38052 rencldnfilem 42831 climlimsupcex 45784 icccncfext 45902 | 
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