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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 2677 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)) | |
2 | 1 | biimpcd 238 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶)) |
3 | 2 | con3dimp 456 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: ssnelpss 3680 opthwiener 4901 ssfin4 9015 pwxpndom2 9366 fzneuz 12290 hauspwpwf1 21601 vdgr1b 26431 topdifinffinlem 32371 clsk1indlem1 37363 |
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