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Theorem nssinpss 3573
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 3561 . . 3  |-  ( A  i^i  B )  C_  A
21biantrur 493 . 2  |-  ( ( A  i^i  B )  =/=  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
3 df-ss 3334 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
43necon3bbii 2632 . 2  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  =/=  A
)
5 df-pss 3336 . 2  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
62, 4, 53bitr4i 269 1  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  C.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    =/= wne 2599    i^i cin 3319    C_ wss 3320    C. wpss 3321
This theorem is referenced by:  fbfinnfr  17873  chrelat2i  23868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-in 3327  df-ss 3334  df-pss 3336
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