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Theorem pssdifn0 3528
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 3526 . . . 4  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
2 eqss 3207 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
32simplbi2 608 . . . 4  |-  ( A 
C_  B  ->  ( B  C_  A  ->  A  =  B ) )
41, 3syl5bir 209 . . 3  |-  ( A 
C_  B  ->  (
( B  \  A
)  =  (/)  ->  A  =  B ) )
54necon3d 2497 . 2  |-  ( A 
C_  B  ->  ( A  =/=  B  ->  ( B  \  A )  =/=  (/) ) )
65imp 418 1  |-  ( ( A  C_  B  /\  A  =/=  B )  -> 
( B  \  A
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459    \ cdif 3162    C_ wss 3165   (/)c0 3468
This theorem is referenced by:  pssdif  3529  tz7.7  4434  domdifsn  6961  inf3lem3  7347  isf32lem6  8000  fclscf  17736  flimfnfcls  17739  lebnumlem1  18475  lebnumlem2  18476  lebnumlem3  18477  ig1peu  19573  ig1pdvds  19578  divrngidl  26756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469
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