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Theorem nssne2 3566
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 3530 . . . 4  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
21biimpcd 224 . . 3  |-  ( A 
C_  C  ->  ( A  =  B  ->  B 
C_  C ) )
32necon3bd 2679 . 2  |-  ( A 
C_  C  ->  ( -.  B  C_  C  ->  A  =/=  B ) )
43imp 429 1  |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    =/= wne 2662    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ne 2664  df-in 3488  df-ss 3495
This theorem is referenced by:  atcvatlem  27127  mdsymlem3  27147  disjdifprg  27259  mapdh6aN  36933  mapdh8e  36982  hdmap1l6a  37008
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