Theorem unssbd 3254

Description: If ( A u. B ) is contained in C, so is B. One-way deduction form of unss 3250. Partial converse of unssd 3252. (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 3250	. . 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 3069   C_ wss 3071

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084

This theorem is referenced by:  eldifpw  4398  ertr  6444  diffifi  6788  sumsplitdc  11201  fsum2dlemstep  11203  fsumabs  11234  fsumiun  11246