Theorem unssd 3218

Description: A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
unssd.1	|- ( ph -> A C_ C )
unssd.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
unssd	|- ( ph -> ( A u. B ) C_ C )

Proof of Theorem unssd

Step	Hyp	Ref	Expression
1		unssd.1	. 2 |- ( ph -> A C_ C )
2		unssd.2	. 2 |- ( ph -> B C_ C )
3		unss 3216	. . 3 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
4	3	biimpi 119	. 2 |- ( ( A C_ C /\ B C_ C ) -> ( A u. B ) C_ C )
5	1, 2, 4	syl2anc 406	1 |- ( ph -> ( A u. B ) C_ C )

Syntax hints:   -> wi 4   /\ wa 103   u. cun 3035   C_ wss 3037

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050

This theorem is referenced by:  tpssi  3652  casef  6925  un0addcl  8914  un0mulcl  8915  fzosplit  9847  fzouzsplit  9849  exmidunben  11784  strleund  11890  fsumcncntop  12542  bj-omtrans  12846