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Theorem nssne2 3407
Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
Assertion
Ref Expression
nssne2  |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B
)

Proof of Theorem nssne2
StepHypRef Expression
1 sseq1 3371 . . . 4  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
21biimpcd 217 . . 3  |-  ( A 
C_  C  ->  ( A  =  B  ->  B 
C_  C ) )
32necon3bd 2640 . 2  |-  ( A 
C_  C  ->  ( -.  B  C_  C  ->  A  =/=  B ) )
43imp 420 1  |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    =/= wne 2601    C_ wss 3322
This theorem is referenced by:  atcvatlem  23890  mdsymlem3  23910  disjdifprg  24019  mapdh6aN  32595  mapdh8e  32644  hdmap1l6a  32670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ne 2603  df-in 3329  df-ss 3336
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