Theorem unssbd 3300

Description: If ( A u. B ) is contained in C, so is B. One-way deduction form of unss 3296. Partial converse of unssd 3298. (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 3296	. . 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 3114   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by:  eldifpw  4455  ertr  6516  diffifi  6860  sumsplitdc  11373  fsum2dlemstep  11375  fsumabs  11406  fsumiun  11418  fprod2dlemstep  11563