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Theorem pssdifn0 3681
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 3678 . . . 4  |-  ( B 
C_  A  <->  ( B  \  A )  =  (/) )
2 eqss 3355 . . . . 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 2636 . 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 1652    =/= wne 2598    \ cdif 3309    C_ wss 3312   (/)c0 3620
This theorem is referenced by:  pssdif  3682  tz7.7  4599  domdifsn  7183  inf3lem3  7577  isf32lem6  8230  fclscf  18049  flimfnfcls  18052  lebnumlem1  18978  lebnumlem2  18979  lebnumlem3  18980  ig1peu  20086  ig1pdvds  20091  divrngidl  26629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621
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