Theorem unssad 3340

Description: If ( A u. B ) is contained in C, so is A. One-way deduction form of unss 3337. Partial converse of unssd 3339. (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 3337	. . 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 3155   C_ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by:  ersym  6604  findcard2d  6952  findcard2sd  6953  diffifi  6955  sumsplitdc  11597  fsumabs  11630  fsumiun  11642