Theorem unss2 3392

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

Step	Hyp	Ref	Expression
1		unss1 3390	. 2 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
2		uncom 3365	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3365	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3sstr4g 3283	1 |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) )

Syntax hints:   -> wi 4   u. cun 3211   C_ wss 3213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-in 3219  df-ss 3226

This theorem is referenced by:  unss12  3393  difdif2ss  3480  difdifdirss  3596  ord3ex  4305  rdgss  6616  xpider  6842