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Theorem pssdifn0 3632
Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
Assertion
Ref Expression
pssdifn0  |-  ( ( A  C_  B  /\  A  =/=  B )  -> 
( B  \  A
)  =/=  (/) )

Proof of Theorem pssdifn0
StepHypRef Expression
1 ssdif0 3629 . . . 4  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
2 eqss 3306 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
32simplbi2 609 . . . 4  |-  ( A 
C_  B  ->  ( B  C_  A  ->  A  =  B ) )
41, 3syl5bir 210 . . 3  |-  ( A 
C_  B  ->  (
( B  \  A
)  =  (/)  ->  A  =  B ) )
54necon3d 2588 . 2  |-  ( A 
C_  B  ->  ( A  =/=  B  ->  ( B  \  A )  =/=  (/) ) )
65imp 419 1  |-  ( ( A  C_  B  /\  A  =/=  B )  -> 
( B  \  A
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    =/= wne 2550    \ cdif 3260    C_ wss 3263   (/)c0 3571
This theorem is referenced by:  pssdif  3633  tz7.7  4548  domdifsn  7127  inf3lem3  7518  isf32lem6  8171  fclscf  17978  flimfnfcls  17981  lebnumlem1  18857  lebnumlem2  18858  lebnumlem3  18859  ig1peu  19961  ig1pdvds  19966  divrngidl  26329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-in 3270  df-ss 3277  df-nul 3572
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