Theorem unssad 3349

Description: If ( A u. B ) is contained in C, so is A. One-way deduction form of unss 3346. Partial converse of unssd 3348. (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 3346	. . 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 3163   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178

This theorem is referenced by:  ersym  6631  findcard2d  6987  findcard2sd  6988  diffifi  6990  sumsplitdc  11714  fsumabs  11747  fsumiun  11759