Theorem unssbd 3387

Description: If ( A u. B ) is contained in C, so is B. One-way deduction form of unss 3383. Partial converse of unssd 3385. (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 3383	. . 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	simprd 114	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   /\ wa 104   u. cun 3199   C_ wss 3201

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214

This theorem is referenced by:  eldifpw  4580  ertr  6760  diffifi  7126  sumsplitdc  12054  fsum2dlemstep  12056  fsumabs  12087  fsumiun  12099  fprod2dlemstep  12244