![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3039 | . 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
2 | nelne2 3032 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵) | |
3 | 1, 2 | sylan2b 593 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2098 ≠ wne 2932 ∉ wnel 3038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-cleq 2716 df-clel 2802 df-ne 2933 df-nel 3039 |
This theorem is referenced by: nelrnfvne 7069 eldmrexrnb 7083 absprodnn 16552 frgrncvvdeqlem2 30022 frgrncvvdeqlem3 30023 afv0nbfvbi 46344 uniimaelsetpreimafv 46549 imasetpreimafvbijlemfv1 46556 2zrngnmlid 47118 2zrngnmrid 47119 |
Copyright terms: Public domain | W3C validator |