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Theorem nss 3249
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 1575 . . 3  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A. x
( x  e.  A  ->  x  e.  B ) )
2 dfss2 3182 . . 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 1530   E.wex 1531    e. wcel 1696    C_ wss 3165
This theorem is referenced by:  grur1  8458  psslinpr  8671  reclem2pr  8688  mreexexlem2d  13563  prmcyg  15196  filcon  17594  alexsubALTlem4  17760  wilthlem2  20323  shne0i  22043  erdszelem10  23746  fundmpss  24193  vxveqv  25157
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-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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