Theorem unssd 3119

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 3117	. . 3 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
4	3	biimpi 113	. 2 |- ( ( A C_ C /\ B C_ C ) -> ( A u. B ) C_ C )
5	1, 2, 4	syl2anc 391	1 |- ( ph -> ( A u. B ) C_ C )

Syntax hints:   -> wi 4   /\ wa 97   u. cun 2915   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931

This theorem is referenced by:  tpssi  3530  un0addcl  8215  un0mulcl  8216  fzosplit  9033  fzouzsplit  9035  bj-omtrans  10081