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Theorem unss2 3542
Description: Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
unss2  |-  ( A 
C_  B  ->  ( C  u.  A )  C_  ( C  u.  B
) )

Proof of Theorem unss2
StepHypRef Expression
1 unss1 3540 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 uncom 3515 . 2  |-  ( C  u.  A )  =  ( A  u.  C
)
3 uncom 3515 . 2  |-  ( C  u.  B )  =  ( B  u.  C
)
41, 2, 33sstr4g 3412 1  |-  ( A 
C_  B  ->  ( C  u.  A )  C_  ( C  u.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    u. cun 3341    C_ wss 3343
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 2577  df-v 2989  df-un 3348  df-in 3350  df-ss 3357
This theorem is referenced by:  unss12  3543  ord3ex  4497  xpider  7186  fin1a2lem13  8596  canthp1lem2  8835  uniioombllem3  21080  volcn  21101  dvres2lem  21400  bnj1413  32045  bnj1408  32046
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