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Theorem nssss 4245
Description: Negation of subclass relationship. Compare nss 3249. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
nssss  |-  ( -.  A  C_  B  <->  E. x
( x  C_  A  /\  -.  x  C_  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem nssss
StepHypRef Expression
1 exanali 1575 . . 3  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4243 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
31, 2xchbinxr 302 . 2  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A  C_  B )
43bicomi 193 1  |-  ( -.  A  C_  B  <->  E. x
( x  C_  A  /\  -.  x  C_  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    C_ wss 3165
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  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-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660
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