Theorem unssbd 3285

Description: If ( A u. B ) is contained in C, so is B. One-way deduction form of unss 3281. Partial converse of unssd 3283. (Contributed by David Moews, 1-May-2017.)

Hypothesis

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

Assertion

Ref	Expression
unssbd	|- ( ph -> B C_ C )

Proof of Theorem unssbd

Step	Hyp	Ref	Expression
1		unssad.1	. . 3 |- ( ph -> ( A u. B ) C_ C )
2		unss 3281	. . 3 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
3	1, 2	sylibr 133	. 2 |- ( ph -> ( A C_ C /\ B C_ C ) )
4	3	simprd 113	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   /\ wa 103   u. cun 3100   C_ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115

This theorem is referenced by:  eldifpw  4437  ertr  6495  diffifi  6839  sumsplitdc  11329  fsum2dlemstep  11331  fsumabs  11362  fsumiun  11374  fprod2dlemstep  11519