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Theorem pssnel 3695
Description: A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
Assertion
Ref Expression
pssnel  |-  ( A 
C.  B  ->  E. x
( x  e.  B  /\  -.  x  e.  A
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem pssnel
StepHypRef Expression
1 pssdif 3692 . . 3  |-  ( A 
C.  B  ->  ( B  \  A )  =/=  (/) )
2 n0 3639 . . 3  |-  ( ( B  \  A )  =/=  (/)  <->  E. x  x  e.  ( B  \  A
) )
31, 2sylib 190 . 2  |-  ( A 
C.  B  ->  E. x  x  e.  ( B  \  A ) )
4 eldif 3332 . . 3  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
54exbii 1593 . 2  |-  ( E. x  x  e.  ( B  \  A )  <->  E. x ( x  e.  B  /\  -.  x  e.  A ) )
63, 5sylib 190 1  |-  ( A 
C.  B  ->  E. x
( x  e.  B  /\  -.  x  e.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   E.wex 1551    e. wcel 1726    =/= wne 2601    \ cdif 3319    C. wpss 3323   (/)c0 3630
This theorem is referenced by:  php  7293  php3  7295  pssnn  7329  inf3lem2  7586  infpssr  8190  ssfin4  8192  genpnnp  8884  ltexprlem1  8915  reclem2pr  8927  mrieqv2d  13866  lbspss  16156  lsmcv  16215  lidlnz  16301  obslbs  16959  nmoid  18778  spansncvi  23156  lsat0cv  29893  osumcllem11N  30825  pexmidlem8N  30836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631
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