Theorem unssd 3312

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 3310	. . 3 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
4	3	biimpi 120	. 2 |- ( ( A C_ C /\ B C_ C ) -> ( A u. B ) C_ C )
5	1, 2, 4	syl2anc 411	1 |- ( ph -> ( A u. B ) C_ C )

Syntax hints:   -> wi 4   /\ wa 104   u. cun 3128   C_ wss 3130

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143

This theorem is referenced by:  tpssi  3760  casef  7087  un0addcl  9209  un0mulcl  9210  fzosplit  10177  fzouzsplit  10179  exmidunben  12427  strleund  12562  fsumcncntop  14059  bj-charfun  14562  bj-omtrans  14711