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Mirrors > Home > ILE Home > Th. List > elnelne2 | GIF version |
Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.) |
Ref | Expression |
---|---|
elnelne2 | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 2404 | . 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
2 | nelne2 2399 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵) | |
3 | 1, 2 | sylan2b 285 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1480 ≠ wne 2308 ∉ wnel 2403 |
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 603 ax-in2 604 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 df-clel 2135 df-ne 2309 df-nel 2404 |
This theorem is referenced by: (None) |
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