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Theorem nss 3366
Description: Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
nss  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem nss
StepHypRef Expression
1 exanali 1592 . . 3  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A. x
( x  e.  A  ->  x  e.  B ) )
2 dfss2 3297 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
31, 2xchbinxr 303 . 2  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A  C_  B )
43bicomi 194 1  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    e. wcel 1721    C_ wss 3280
This theorem is referenced by:  grur1  8651  psslinpr  8864  reclem2pr  8881  mreexexlem2d  13825  prmcyg  15458  filcon  17868  alexsubALTlem4  18034  wilthlem2  20805  shne0i  22903  erdszelem10  24839  fundmpss  25336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-in 3287  df-ss 3294
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