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Theorem unss2 3348
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 3346 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 uncom 3321 . 2  |-  ( C  u.  A )  =  ( A  u.  C
)
3 uncom 3321 . 2  |-  ( C  u.  B )  =  ( B  u.  C
)
41, 2, 33sstr4g 3221 1  |-  ( A 
C_  B  ->  ( C  u.  A )  C_  ( C  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    u. cun 3152    C_ wss 3154
This theorem is referenced by:  unss12  3349  ord3ex  4202  xpider  6732  fin1a2lem13  8040  canthp1lem2  8277  uniioombllem3  18942  volcn  18963  dvres2lem  19262  bnj1413  29138  bnj1408  29139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-un 3159  df-in 3161  df-ss 3168
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