Theorem unssbd 3150

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3146. Partial converse of unssd 3148. (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 3146	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
3	1, 2	sylibr 132	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
4	3	simprd 112	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 102   ∪ cun 2971   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by:  eldifpw  4226  ertr  6144  diffifi  6378