Theorem elnelne2 2505

Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)

Assertion

Ref	Expression
elnelne2	|- ( ( A e. C /\ B e/ C ) -> A =/= B )

Proof of Theorem elnelne2

Step	Hyp	Ref	Expression
1		df-nel 2496	. 2 |- ( B e/ C <-> -. B e. C )
2		nelne2 2491	. 2 |- ( ( A e. C /\ -. B e. C ) -> A =/= B )
3	1, 2	sylan2b 287	1 |- ( ( A e. C /\ B e/ C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2200   =/= wne 2400   e/ wnel 2495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-ne 2401  df-nel 2496

This theorem is referenced by: (None)