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Theorem nsspssun 3534
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 3471 . . . 4  |-  B  C_  ( A  u.  B
)
21biantrur 493 . . 3  |-  ( -.  ( A  u.  B
)  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
3 ssid 3327 . . . . 5  |-  B  C_  B
43biantru 492 . . . 4  |-  ( A 
C_  B  <->  ( A  C_  B  /\  B  C_  B ) )
5 unss 3481 . . . 4  |-  ( ( A  C_  B  /\  B  C_  B )  <->  ( A  u.  B )  C_  B
)
64, 5bitri 241 . . 3  |-  ( A 
C_  B  <->  ( A  u.  B )  C_  B
)
72, 6xchnxbir 301 . 2  |-  ( -.  A  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
8 dfpss3 3393 . 2  |-  ( B 
C.  ( A  u.  B )  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
97, 8bitr4i 244 1  |-  ( -.  A  C_  B  <->  B  C.  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    u. cun 3278    C_ wss 3280    C. wpss 3281
This theorem is referenced by:  disjpss  3638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296
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