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Mirrors > Home > ILE Home > Th. List > nelneq2 | 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 2253 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)) | |
2 | 1 | biimpcd 159 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶)) |
3 | 2 | con3dimp 636 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-clel 2185 |
This theorem is referenced by: dtruarb 4209 fzneuz 10133 |
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