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Theorem nssss 4703
Description: Negation of subclass relationship. Compare nss 3562. (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 1647 . . 3  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4701 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
31, 2xchbinxr 311 . 2  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  <->  -.  A  C_  B )
43bicomi 202 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 184    /\ wa 369   A.wal 1377   E.wex 1596    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030
This theorem is referenced by: (None)
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