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Mirrors > Home > MPE Home > Th. List > nelelne | Structured version Visualization version GIF version |
Description: Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.) |
Ref | Expression |
---|---|
nelelne | ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelne2 3115 | . 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵) → 𝐶 ≠ 𝐴) | |
2 | 1 | expcom 414 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2105 ≠ wne 3016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2814 df-clel 2893 df-ne 3017 |
This theorem is referenced by: difsn 4725 elneq 9051 frgrncvvdeqlem7 28012 frgrncvvdeqlem9 28014 prter2 35899 |
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