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Theorem pssdif 3529
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 3181 . 2  |-  ( A 
C.  B  <->  ( A  C_  B  /\  A  =/= 
B ) )
2 pssdifn0 3528 . 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 2459    \ cdif 3162    C_ wss 3165    C. wpss 3166   (/)c0 3468
This theorem is referenced by:  pssnel  3532  pgpfac1lem5  15330  fundmpss  24193  dfon2lem6  24215
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-pss 3181  df-nul 3469
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