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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 2901 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)) | |
2 | 1 | biimpcd 251 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶)) |
3 | 2 | con3dimp 411 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-cleq 2814 df-clel 2893 |
This theorem is referenced by: nelne1 3113 ssnelpss 4087 opthwiener 5403 ssfin4 9731 pwxpndom2 10086 fzneuz 12987 hauspwpwf1 22594 topdifinffinlem 34627 clsk1indlem1 40393 |
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