Theorem unss12 3376

Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)

Assertion

Ref	Expression
unss12	|- ( ( A C_ B /\ C C_ D ) -> ( A u. C ) C_ ( B u. D ) )

Proof of Theorem unss12

Step	Hyp	Ref	Expression
1		unss1 3373	. 2 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
2		unss2 3375	. 2 |- ( C C_ D -> ( B u. C ) C_ ( B u. D ) )
3	1, 2	sylan9ss 3237	1 |- ( ( A C_ B /\ C C_ D ) -> ( A u. C ) C_ ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 104   u. cun 3195   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  ifssun  3617  fun  5496  resasplitss  5504  lspun  14360