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Theorem nssinpss 3641
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 3618 . . 3  |-  ( A  i^i  B )  C_  A
21biantrur 508 . 2  |-  ( ( A  i^i  B )  =/=  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
3 df-ss 3386 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
43necon3bbii 2642 . 2  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  =/=  A
)
5 df-pss 3388 . 2  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
62, 4, 53bitr4i 280 1  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  C.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    =/= wne 2593    i^i cin 3371    C_ wss 3372    C. wpss 3373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-v 3018  df-in 3379  df-ss 3386  df-pss 3388
This theorem is referenced by:  fbfinnfr  20791  chrelat2i  27953
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