Theorem elnelne2 2472

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 2463	. 2 |- ( B e/ C <-> -. B e. C )
2		nelne2 2458	. 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 2167   =/= wne 2367   e/ wnel 2462

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-ne 2368  df-nel 2463

This theorem is referenced by: (None)