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Theorem pssnel 3519
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 3516 . . 3  |-  ( A 
C.  B  ->  ( B  \  A )  =/=  (/) )
2 n0 3464 . . 3  |-  ( ( B  \  A )  =/=  (/)  <->  E. x  x  e.  ( B  \  A
) )
31, 2sylib 188 . 2  |-  ( A 
C.  B  ->  E. x  x  e.  ( B  \  A ) )
4 eldif 3162 . . 3  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
54exbii 1569 . 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 1528    e. wcel 1684    =/= wne 2446    \ cdif 3149    C. wpss 3153   (/)c0 3455
This theorem is referenced by:  php  7045  php3  7047  pssnn  7081  inf3lem2  7330  infpssr  7934  ssfin4  7936  genpnnp  8629  ltexprlem1  8660  reclem2pr  8672  mrieqv2d  13541  lbspss  15835  lsmcv  15894  lidlnz  15980  obslbs  16630  nmoid  18251  spansncvi  22231  lsat0cv  29223  osumcllem11N  30155  pexmidlem8N  30166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456
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