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Metamath Proof Explorer

Theorem iunssd

Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021)

		Ref	Expression
	Hypothesis	iunssd.1	$⊢ (φ \land x \in A) \to B \subseteq C$
	Assertion	iunssd	$⊢ φ \to ⋃_{x \in A} B \subseteq C$

Step	Hyp	Ref	Expression
1		iunssd.1	$⊢ (φ \land x \in A) \to B \subseteq C$
2	1	ralrimiva	$⊢ φ \to \forall x \in A B \subseteq C$
3		iunss	$⊢ ⋃_{x \in A} B \subseteq C \leftrightarrow \forall x \in A B \subseteq C$
4	2 3	sylibr	$⊢ φ \to ⋃_{x \in A} B \subseteq C$