Theorem unssbd

Description: If ( A u. B ) is contained in C , so is B . One-way deduction form of unss . Partial converse of unssd . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypothesis	unssad.1	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ⊆ 𝐶 )
Assertion	unssbd	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )

Proof

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