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Theorem nssss 4678
Description: Negation of subclass relationship. Compare nss 3528. (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 1717 . . 3  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4676 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
31, 2xchbinxr 312 . 2  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A  C_  B )
43bicomi 205 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 187    /\ wa 370   A.wal 1435   E.wex 1659    C_ wss 3442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-pw 3987  df-sn 4003  df-pr 4005
This theorem is referenced by: (None)
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