Theorem unss2 3352

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 3350	. 2 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
2		uncom 3325	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3325	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3sstr4g 3244	1 |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) )

Syntax hints:   -> wi 4   u. cun 3172   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187

This theorem is referenced by:  unss12  3353  difdif2ss  3438  difdifdirss  3553  ord3ex  4250  rdgss  6492  xpider  6716