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Theorem nssinpss 3166
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 3154 . . 3  |-  ( A  i^i  B )  C_  A
21biantrur 287 . 2  |-  ( ( A  i^i  B )  =/=  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
3 df-ss 2928 . . 3  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
43necon3bbii 2239 . 2  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  =/=  A
)
5 df-pss 2930 . 2  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
62, 4, 53bitr4i 201 1  |-  ( -.  A  C_  B  <->  ( A  i^i  B )  C.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 97    <-> wb 98    =/= wne 2204    i^i cin 2913    C_ wss 2914    C. wpss 2915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2556  df-in 2921  df-ss 2928  df-pss 2930
This theorem is referenced by: (None)
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