Theorem elnelne1 2413

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 2405	. 2 |- ( A e/ C <-> -. A e. C )
2		nelne1 2399	. 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 1481   =/= wne 2309   e/ wnel 2404

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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136  df-ne 2310  df-nel 2405

This theorem is referenced by: (None)