Theorem unssd 3247

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 3245	. . 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 408	1 |- ( ph -> ( A u. B ) C_ C )

Syntax hints:   -> wi 4   /\ wa 103   u. cun 3064   C_ wss 3066

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079

This theorem is referenced by:  tpssi  3681  casef  6966  un0addcl  9003  un0mulcl  9004  fzosplit  9947  fzouzsplit  9949  exmidunben  11928  strleund  12036  fsumcncntop  12714  bj-omtrans  13143