Theorem elnelne1 2451

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 2443	. 2 |- ( A e/ C <-> -. A e. C )
2		nelne1 2437	. 2 |- ( ( A e. B /\ -. A e. C ) -> B =/= C )
3	1, 2	sylan2b 287	1 |- ( ( A e. B /\ A e/ C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2148   =/= wne 2347   e/ wnel 2442

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-ne 2348  df-nel 2443

This theorem is referenced by: (None)