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Theorem unss2 3359
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 3357 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 uncom 3332 . 2  |-  ( C  u.  A )  =  ( A  u.  C
)
3 uncom 3332 . 2  |-  ( C  u.  B )  =  ( B  u.  C
)
41, 2, 33sstr4g 3232 1  |-  ( A 
C_  B  ->  ( C  u.  A )  C_  ( C  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    u. cun 3163    C_ wss 3165
This theorem is referenced by:  unss12  3360  ord3ex  4216  xpider  6746  fin1a2lem13  8054  canthp1lem2  8291  uniioombllem3  18956  volcn  18977  dvres2lem  19276  bnj1413  29381  bnj1408  29382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179
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