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Theorem nsspssun 3576
Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
nsspssun  |-  ( -.  A  C_  B  <->  B  C.  ( A  u.  B
) )

Proof of Theorem nsspssun
StepHypRef Expression
1 ssun2 3513 . . . 4  |-  B  C_  ( A  u.  B
)
21biantrur 494 . . 3  |-  ( -.  ( A  u.  B
)  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
3 ssid 3369 . . . . 5  |-  B  C_  B
43biantru 493 . . . 4  |-  ( A 
C_  B  <->  ( A  C_  B  /\  B  C_  B ) )
5 unss 3523 . . . 4  |-  ( ( A  C_  B  /\  B  C_  B )  <->  ( A  u.  B )  C_  B
)
64, 5bitri 242 . . 3  |-  ( A 
C_  B  <->  ( A  u.  B )  C_  B
)
72, 6xchnxbir 302 . 2  |-  ( -.  A  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
8 dfpss3 3435 . 2  |-  ( B 
C.  ( A  u.  B )  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
97, 8bitr4i 245 1  |-  ( -.  A  C_  B  <->  B  C.  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360    u. cun 3320    C_ wss 3322    C. wpss 3323
This theorem is referenced by:  disjpss  3680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338
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