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Theorem nsspssun 3728
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 3654 . . . 4  |-  B  C_  ( A  u.  B
)
21biantrur 504 . . 3  |-  ( -.  ( A  u.  B
)  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
3 ssid 3508 . . . . 5  |-  B  C_  B
43biantru 503 . . . 4  |-  ( A 
C_  B  <->  ( A  C_  B  /\  B  C_  B ) )
5 unss 3664 . . . 4  |-  ( ( A  C_  B  /\  B  C_  B )  <->  ( A  u.  B )  C_  B
)
64, 5bitri 249 . . 3  |-  ( A 
C_  B  <->  ( A  u.  B )  C_  B
)
72, 6xchnxbir 307 . 2  |-  ( -.  A  C_  B  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
8 dfpss3 3576 . 2  |-  ( B 
C.  ( A  u.  B )  <->  ( B  C_  ( A  u.  B
)  /\  -.  ( A  u.  B )  C_  B ) )
97, 8bitr4i 252 1  |-  ( -.  A  C_  B  <->  B  C.  ( A  u.  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    u. cun 3459    C_ wss 3461    C. wpss 3462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477
This theorem is referenced by:  disjpss  3865
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