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| Mirrors > Home > MPE Home > Th. List > nelneq2 | Structured version Visualization version GIF version | ||
| Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.) |
| Ref | Expression |
|---|---|
| nelneq2 | ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq2 2830 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)) | |
| 2 | 1 | biimpcd 249 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶)) |
| 3 | 2 | con3dimp 408 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) |
| 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: nelne1 3039 ssnelpss 4114 opthwiener 5519 ssfin4 10350 pwxpndom2 10705 fzneuz 13648 hauspwpwf1 23995 topdifinffinlem 37348 clsk1indlem1 44058 |
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