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Theorem elnelne2 3139
Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
Assertion
Ref Expression
elnelne2 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)

Proof of Theorem elnelne2
StepHypRef Expression
1 df-nel 3129 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 3120 . 2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
31, 2sylan2b 593 1 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2107  wne 3021  wnel 3128
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2819  df-clel 2898  df-ne 3022  df-nel 3129
This theorem is referenced by:  nelrnfvne  6841  eldmrexrnb  6854  absprodnn  15952  frgrncvvdeqlem2  27993  frgrncvvdeqlem3  27994  afv0nbfvbi  43216  2zrngnmlid  44052  2zrngnmrid  44053
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