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Theorem nss 3238
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 1574 . . 3  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A. x
( x  e.  A  ->  x  e.  B ) )
2 dfss2 3171 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
31, 2xchbinxr 302 . 2  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A  C_  B )
43bicomi 193 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 176    /\ wa 358   A.wal 1529   E.wex 1530    e. wcel 1686    C_ wss 3154
This theorem is referenced by:  grur1  8444  psslinpr  8657  reclem2pr  8674  mreexexlem2d  13549  prmcyg  15182  filcon  17580  alexsubALTlem4  17746  wilthlem2  20309  shne0i  22029  erdszelem10  23733  fundmpss  24124  vxveqv  25065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-in 3161  df-ss 3168
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