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Theorem nssinpss 3737
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 3714 . . 3  |-  ( A  i^i  B )  C_  A
21biantrur 506 . 2  |-  ( ( A  i^i  B )  =/=  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
3 df-ss 3485 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
43necon3bbii 2718 . 2  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  =/=  A
)
5 df-pss 3487 . 2  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
62, 4, 53bitr4i 277 1  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  C.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    =/= wne 2652    i^i cin 3470    C_ wss 3471    C. wpss 3472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-in 3478  df-ss 3485  df-pss 3487
This theorem is referenced by:  fbfinnfr  20467  chrelat2i  27410
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