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Mirrors > Home > ILE Home > Th. List > nelneq | 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 2141 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | 1 | biimpcd 157 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶)) |
3 | 2 | con3dimp 596 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-clel 2077 |
This theorem is referenced by: (None) |
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