Theorem elnelne1 2444

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

Assertion

Ref	Expression
elnelne1	|- ( ( A e. B /\ A e/ C ) -> B =/= C )

Proof of Theorem elnelne1

Step	Hyp	Ref	Expression
1		df-nel 2436	. 2 |- ( A e/ C <-> -. A e. C )
2		nelne1 2430	. 2 |- ( ( A e. B /\ -. A e. C ) -> B =/= C )
3	1, 2	sylan2b 285	1 |- ( ( A e. B /\ A e/ C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2141   =/= wne 2340   e/ wnel 2435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-ne 2341  df-nel 2436

This theorem is referenced by: (None)