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Mirrors > Home > ILE Home > Th. List > nelne2 | GIF version |
Description: Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.) |
Ref | Expression |
---|---|
nelne2 | ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2227 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | 1 | biimpcd 158 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶)) |
3 | 2 | necon3bd 2377 | . 2 ⊢ (𝐴 ∈ 𝐶 → (¬ 𝐵 ∈ 𝐶 → 𝐴 ≠ 𝐵)) |
4 | 3 | imp 123 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-cleq 2157 df-clel 2160 df-ne 2335 |
This theorem is referenced by: nelelne 2426 elnelne2 2439 zgt1rpn0n1 9622 |
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