Theorem elnelne2 2908

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 2898	. 2 |- ( B e/ C <-> -. B e. C )
2		nelne2 2891	. 2 |- ( ( A e. C /\ -. B e. C ) -> A =/= B )
3	1, 2	sylan2b 492	1 |- ( ( A e. C /\ B e/ C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   e/ wnel 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ne 2795  df-nel 2898

This theorem is referenced by:  nelrnfvne  6353  eldmrexrnb  6366  absprodnn  15331  frgrncvvdeqlem2  27164  frgrncvvdeqlem3  27165  afv0nbfvbi  41231  2zrngnmlid  41949  2zrngnmrid  41950