Theorem bnj226

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj226.1	$$⊢B\subseteq C$$
Assertion	bnj226	$$⊢\bigcup _{x\in A}B\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		bnj226.1	$$⊢B\subseteq C$$
2	1	rgenw	$$⊢\forall x\in AB\subseteq C$$
3		iunss	$$⊢\bigcup _{x\in A}B\subseteq C\leftrightarrow \forall x\in AB\subseteq C$$
4	2 3	mpbir	$$⊢\bigcup _{x\in A}B\subseteq C$$