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Theorem pssdif 3626
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 3272 . 2  |-  ( A 
C.  B  <->  ( A  C_  B  /\  A  =/= 
B ) )
2 pssdifn0 3625 . 2  |-  ( ( A  C_  B  /\  A  =/=  B )  -> 
( B  \  A
)  =/=  (/) )
31, 2sylbi 188 1  |-  ( A 
C.  B  ->  ( B  \  A )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    =/= wne 2543    \ cdif 3253    C_ wss 3256    C. wpss 3257   (/)c0 3564
This theorem is referenced by:  pssnel  3629  pgpfac1lem5  15557  fundmpss  25139  dfon2lem6  25161
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 2361
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-v 2894  df-dif 3259  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565
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