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Theorem nsspssun 3402
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 3339 . . . 4  |-  B  C_  ( A  u.  B
)
21biantrur 492 . . 3  |-  ( -.  ( A  u.  B
)  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
3 ssid 3197 . . . . 5  |-  B  C_  B
43biantru 491 . . . 4  |-  ( A 
C_  B  <->  ( A  C_  B  /\  B  C_  B ) )
5 unss 3349 . . . 4  |-  ( ( A  C_  B  /\  B  C_  B )  <->  ( A  u.  B )  C_  B
)
64, 5bitri 240 . . 3  |-  ( A 
C_  B  <->  ( A  u.  B )  C_  B
)
72, 6xchnxbir 300 . 2  |-  ( -.  A  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
8 dfpss3 3262 . 2  |-  ( B 
C.  ( A  u.  B )  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
97, 8bitr4i 243 1  |-  ( -.  A  C_  B  <->  B  C.  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    u. cun 3150    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  disjpss  3505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168
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