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Theorem pssnel 3532
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 3529 . . 3  |-  ( A 
C.  B  ->  ( B  \  A )  =/=  (/) )
2 n0 3477 . . 3  |-  ( ( B  \  A )  =/=  (/)  <->  E. x  x  e.  ( B  \  A
) )
31, 2sylib 188 . 2  |-  ( A 
C.  B  ->  E. x  x  e.  ( B  \  A ) )
4 eldif 3175 . . 3  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
54exbii 1572 . 2  |-  ( E. x  x  e.  ( B  \  A )  <->  E. x ( x  e.  B  /\  -.  x  e.  A ) )
63, 5sylib 188 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 358   E.wex 1531    e. wcel 1696    =/= wne 2459    \ cdif 3162    C. wpss 3166   (/)c0 3468
This theorem is referenced by:  php  7061  php3  7063  pssnn  7097  inf3lem2  7346  infpssr  7950  ssfin4  7952  genpnnp  8645  ltexprlem1  8676  reclem2pr  8688  mrieqv2d  13557  lbspss  15851  lsmcv  15910  lidlnz  15996  obslbs  16646  nmoid  18267  spansncvi  22247  lsat0cv  29845  osumcllem11N  30777  pexmidlem8N  30788
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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