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Theorem nssss 4655
Description: Negation of subclass relationship. Compare nss 3489. (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 1720 . . 3  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4653 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
31, 2xchbinxr 313 . 2  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A  C_  B )
43bicomi 206 1  |-  ( -.  A  C_  B  <->  E. x
( x  C_  A  /\  -.  x  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1441   E.wex 1662    C_ wss 3403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-pw 3952  df-sn 3968  df-pr 3970
This theorem is referenced by: (None)
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