Theorem unssd 3303

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 3301	. . 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 409	1 |- ( ph -> ( A u. B ) C_ C )

Syntax hints:   -> wi 4   /\ wa 103   u. cun 3119   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  tpssi  3746  casef  7065  un0addcl  9168  un0mulcl  9169  fzosplit  10133  fzouzsplit  10135  exmidunben  12381  strleund  12506  fsumcncntop  13350  bj-charfun  13842  bj-omtrans  13991