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Metamath Proof Explorer

Theorem pwssb

Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006)

		Ref	Expression
	Assertion	pwssb	$⊢ A \subseteq 𝒫 B \leftrightarrow \forall x \in A x \subseteq B$

Step	Hyp	Ref	Expression
1		sspwuni	$⊢ A \subseteq 𝒫 B \leftrightarrow ⋃ A \subseteq B$
2		unissb	$⊢ ⋃ A \subseteq B \leftrightarrow \forall x \in A x \subseteq B$
3	1 2	bitri	$⊢ A \subseteq 𝒫 B \leftrightarrow \forall x \in A x \subseteq B$