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Theorem nss 3211
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 1583 . . 3  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A. x
( x  e.  A  ->  x  e.  B ) )
2 dfss2 3144 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
31, 2xchbinxr 304 . 2  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A  C_  B )
43bicomi 195 1  |-  ( -.  A  C_  B  <->  E. x
( x  e.  A  /\  -.  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537    e. wcel 1621    C_ wss 3127
This theorem is referenced by:  grur1  8410  psslinpr  8623  reclem2pr  8640  mreexexlem2d  13510  prmcyg  15143  filcon  17541  alexsubALTlem4  17707  wilthlem2  20270  shne0i  21988  erdszelem10  23104  fundmpss  23492  vxveqv  24421
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-in 3134  df-ss 3141
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