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	$$⊢\phi \to A\cup B\subseteq C$$
Assertion	unssbd	$$⊢\phi \to B\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		unssad.1	$$⊢\phi \to A\cup B\subseteq C$$
2		unss	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A\cup B\subseteq C$$
3	1 2	sylibr	$$⊢\phi \to \left(A\subseteq C\wedge B\subseteq C\right)$$
4	3	simprd	$$⊢\phi \to B\subseteq C$$