Theorem unss2 3334

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 3332	. 2 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
2		uncom 3307	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3307	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3sstr4g 3226	1 |- ( A C_ B -> ( C u. A ) C_ ( C u. B ) )

Syntax hints:   -> wi 4   u. cun 3155   C_ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by:  unss12  3335  difdif2ss  3420  difdifdirss  3535  ord3ex  4223  rdgss  6441  xpider  6665