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Theorem nssinpss 3401
Description: Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
nssinpss  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  C.  A
)

Proof of Theorem nssinpss
StepHypRef Expression
1 inss1 3389 . . 3  |-  ( A  i^i  B )  C_  A
21biantrur 492 . 2  |-  ( ( A  i^i  B )  =/=  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
3 df-ss 3166 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
43necon3bbii 2477 . 2  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  =/=  A
)
5 df-pss 3168 . 2  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
62, 4, 53bitr4i 268 1  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  C.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    =/= wne 2446    i^i cin 3151    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  fbfinnfr  17536  chrelat2i  22945
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-in 3159  df-ss 3166  df-pss 3168
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