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Theorem pssdif 3682
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 3328 . 2  |-  ( A 
C.  B  <->  ( A  C_  B  /\  A  =/= 
B ) )
2 pssdifn0 3681 . 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 2598    \ cdif 3309    C_ wss 3312    C. wpss 3313   (/)c0 3620
This theorem is referenced by:  pssnel  3685  pgpfac1lem5  15629  fundmpss  25382  dfon2lem6  25407
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-pss 3328  df-nul 3621
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