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Theorem unss12 3528
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 3525 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 unss2 3527 . 2  |-  ( C 
C_  D  ->  ( B  u.  C )  C_  ( B  u.  D
) )
31, 2sylan9ss 3369 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 3326    C_ wss 3328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-un 3333  df-in 3335  df-ss 3342
This theorem is referenced by:  pwssun  4627  fun  5575  undom  7399  finsschain  7618  mvdco  15951  dprd2da  16541  dmdprdsplit2lem  16544  lspun  17068  spanuni  24947  sshhococi  24949  mblfinlem3  28430  dochdmj1  35035
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