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Theorem pssdif 3516
Description: A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
Assertion
Ref Expression
pssdif  |-  ( A 
C.  B  ->  ( B  \  A )  =/=  (/) )

Proof of Theorem pssdif
StepHypRef Expression
1 df-pss 3168 . 2  |-  ( A 
C.  B  <->  ( A  C_  B  /\  A  =/= 
B ) )
2 pssdifn0 3515 . 2  |-  ( ( A  C_  B  /\  A  =/=  B )  -> 
( B  \  A
)  =/=  (/) )
31, 2sylbi 187 1  |-  ( A 
C.  B  ->  ( B  \  A )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    =/= wne 2446    \ cdif 3149    C_ wss 3152    C. wpss 3153   (/)c0 3455
This theorem is referenced by:  pssnel  3519  pgpfac1lem5  15314  fundmpss  24122  dfon2lem6  24144
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-pss 3168  df-nul 3456
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