Theorem unssad 3631

Description: If ( A u. B ) is contained in C, so is A. One-way deduction form of unss 3628. Partial converse of unssd 3630. (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 3628	. . 3 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
3	1, 2	sylibr 212	. 2 |- ( ph -> ( A C_ C /\ B C_ C ) )
4	3	simpld 459	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 369   u. cun 3424   C_ wss 3426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3070  df-un 3431  df-in 3433  df-ss 3440

This theorem is referenced by:  ersym  7213  findcard2d  7655  finsschain  7719  r0weon  8280  ackbij1lem16  8505  wunex2  9006  sumsplit  13337  fsumabs  13366  fsumiun  13386  mrieqvlemd  14669  yonedalem1  15184  yonedalem21  15185  yonedalem22  15190  yonffthlem  15194  lsmsp  17273  mplcoe1  17651  mdetunilem9  18542  ordtbas  18912  isufil2  19597  ufileu  19608  filufint  19609  fmfnfm  19647  flimclslem  19673  fclsfnflim  19716  flimfnfcls  19717  imasdsf1olem  20064  mbfeqalem  21236  limcdif  21467  jensenlem1  22496  jensenlem2  22497  jensen  22498  ordtconlem1  26488  rngunsnply  29668