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Theorem unss2 3657
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 3655 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 uncom 3630 . 2  |-  ( C  u.  A )  =  ( A  u.  C
)
3 uncom 3630 . 2  |-  ( C  u.  B )  =  ( B  u.  C
)
41, 2, 33sstr4g 3527 1  |-  ( A 
C_  B  ->  ( C  u.  A )  C_  ( C  u.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    u. cun 3456    C_ wss 3458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-un 3463  df-in 3465  df-ss 3472
This theorem is referenced by:  unss12  3658  ord3ex  4623  xpider  7380  fin1a2lem13  8790  canthp1lem2  9029  uniioombllem3  21860  volcn  21881  dvres2lem  22180  bnj1413  33798  bnj1408  33799
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