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Theorem unss12 3681
Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
Assertion
Ref Expression
unss12  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  u.  C
)  C_  ( B  u.  D ) )

Proof of Theorem unss12
StepHypRef Expression
1 unss1 3678 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 unss2 3680 . 2  |-  ( C 
C_  D  ->  ( B  u.  C )  C_  ( B  u.  D
) )
31, 2sylan9ss 3522 1  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  u.  C
)  C_  ( B  u.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    u. cun 3479    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-in 3488  df-ss 3495
This theorem is referenced by:  pwssun  4792  fun  5754  undom  7617  finsschain  7839  mvdco  16343  dprd2da  16963  dmdprdsplit2lem  16966  lspun  17504  spanuni  26285  sshhococi  26287  mthmpps  28767  mblfinlem3  29980  limciccioolb  31486  limcicciooub  31502  dochdmj1  36588
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