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Theorem ssun 3621
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 3607 . 2  |-  ( A 
C_  B  ->  A  C_  ( B  u.  C
) )
2 ssun4 3608 . 2  |-  ( A 
C_  C  ->  A  C_  ( B  u.  C
) )
31, 2jaoi 377 1  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    u. cun 3411    C_ wss 3413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-un 3418  df-in 3420  df-ss 3427
This theorem is referenced by:  pwunss  4727  pwssun  4728  ordssun  4920  padct  27872
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