Theorem unssbd 3442

Description: If (A ∪ 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	⊢ (φ → (A ∪ B) ⊆ C)

Assertion

Ref	Expression
unssbd	⊢ (φ → B ⊆ C)

Proof of Theorem unssbd

Step	Hyp	Ref	Expression
1		unssad.1	. . 3 ⊢ (φ → (A ∪ B) ⊆ C)
2		unss 3438	. . 3 ⊢ ((A ⊆ C ∧ B ⊆ C) ↔ (A ∪ B) ⊆ C)
3	1, 2	sylibr 203	. 2 ⊢ (φ → (A ⊆ C ∧ B ⊆ C))
4	3	simprd 449	1 ⊢ (φ → B ⊆ C)

Syntax hints:   → wi 4   ∧ wa 358   ∪ cun 3208   ⊆ 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)