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Theorem elnelne2 2802
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
StepHypRef Expression
1 df-nel 2652 . 2  |-  ( B  e/  C  <->  -.  B  e.  C )
2 nelne2 2784 . 2  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  A  =/=  B )
31, 2sylan2b 473 1  |-  ( ( A  e.  C  /\  B  e/  C )  ->  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    e. wcel 1823    =/= wne 2649    e/ wnel 2650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-cleq 2446  df-clel 2449  df-ne 2651  df-nel 2652
This theorem is referenced by:  nelrnfvne  6001  eldmrexrnb  6014  afv0nbfvbi  32475  2zrngnmlid  33009  2zrngnmrid  33010
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