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Theorem pssdifn0 3515
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 3513 . . . 4  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
2 eqss 3194 . . . . 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 2484 . 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 1623    =/= wne 2446    \ cdif 3149    C_ wss 3152   (/)c0 3455
This theorem is referenced by:  pssdif  3516  tz7.7  4418  domdifsn  6945  inf3lem3  7331  isf32lem6  7984  fclscf  17720  flimfnfcls  17723  lebnumlem1  18459  lebnumlem2  18460  lebnumlem3  18461  ig1peu  19557  ig1pdvds  19562  divrngidl  26653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456
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