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| Mirrors > Home > MPE Home > Th. List > elnelne2 | Structured version Visualization version 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 3047 | . 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
| 2 | nelne2 3040 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵) | |
| 3 | 1, 2 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 | 
| This theorem is referenced by: nelrnfvne 7097 eldmrexrnb 7112 absprodnn 16655 frgrncvvdeqlem2 30319 frgrncvvdeqlem3 30320 afv0nbfvbi 47163 uniimaelsetpreimafv 47383 imasetpreimafvbijlemfv1 47390 2zrngnmlid 48171 2zrngnmrid 48172 | 
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