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Theorem nssss 4229
Description: Negation of subclass relationship. Compare nss 3236. (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 1572 . . 3  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4227 . . 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 1527   E.wex 1528    C_ wss 3152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647
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