Theorem unssbd 3442

Description: If (A u. B) is contained in C, so is B. One-way deduction form of unss 3438. Partial converse of unssd 3440. (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 3438	. . 3 |- ((A C_ C /\ B C_ C) <-> (A u. B) C_ C)
3	1, 2	sylibr 203	. 2 |- (ph -> (A C_ C /\ B C_ C))
4	3	simprd 449	1 |- (ph -> B C_ C)

Syntax hints:   -> wi 4   /\ wa 358   u. cun 3208   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by: (None)