Theorem unss2 3293

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 3291	. 2 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
2		uncom 3266	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3266	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3sstr4g 3185	1 |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) )

Syntax hints:   -> wi 4   u. cun 3114   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by:  unss12  3294  difdif2ss  3379  difdifdirss  3493  ord3ex  4169  rdgss  6351  xpider  6572