Theorem unssad 3381

Description: If ( A u. B ) is contained in C, so is A. One-way deduction form of unss 3378. Partial converse of unssd 3380. (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 3378	. . 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 3195   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  ersym  6690  findcard2d  7049  findcard2sd  7050  diffifi  7052  sumsplitdc  11938  fsumabs  11971  fsumiun  11983