MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elnelne2 Structured version   Visualization version   GIF version

Theorem elnelne2 3134
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 3124 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 3115 . 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 2105  wne 3016  wnel 3123
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  df-nel 3124
This theorem is referenced by:  nelrnfvne  6838  eldmrexrnb  6851  absprodnn  15952  frgrncvvdeqlem2  28007  frgrncvvdeqlem3  28008  afv0nbfvbi  43231  2zrngnmlid  44118  2zrngnmrid  44119
  Copyright terms: Public domain W3C validator