Theorem unssad 3384

Description: If ( A u. B ) is contained in C, so is A. One-way deduction form of unss 3381. Partial converse of unssd 3383. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
unssad.1	|- ( ph -> ( A u. B ) C_ C )

Assertion

Ref	Expression
unssad	|- ( ph -> A C_ C )

Proof of Theorem unssad

Step	Hyp	Ref	Expression
1		unssad.1	. . 3 |- ( ph -> ( A u. B ) C_ C )
2		unss 3381	. . 3 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
3	1, 2	sylibr 134	. 2 |- ( ph -> ( A C_ C /\ B C_ C ) )
4	3	simpld 112	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 104   u. cun 3198   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by:  ersym  6713  findcard2d  7079  findcard2sd  7080  diffifi  7082  sumsplitdc  11992  fsumabs  12025  fsumiun  12037