Theorem unss2 3400

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 3398	. 2 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
2		uncom 3373	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3373	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3sstr4g 3291	1 |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) )

Syntax hints:   -> wi 4   u. cun 3218   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by:  unss12  3401  difdif2ss  3488  difdifdirss  3609  ord3ex  4322  rdgss  6644  xpider  6870