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Theorem ssun 3646
Description: A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
ssun  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )

Proof of Theorem ssun
StepHypRef Expression
1 ssun3 3632 . 2  |-  ( A 
C_  B  ->  A  C_  ( B  u.  C
) )
2 ssun4 3633 . 2  |-  ( A 
C_  C  ->  A  C_  ( B  u.  C
) )
31, 2jaoi 379 1  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    u. cun 3437    C_ wss 3439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-in 3446  df-ss 3453
This theorem is referenced by:  pwunss  4737  pwssun  4738  ordssun  4929
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