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Metamath Proof Explorer

Theorem ssintab

Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006) (Proof shortened by Andrew Salmon, 9-Jul-2011)

		Ref	Expression
	Assertion	ssintab	$⊢ A \subseteq ⋂ \{x \| φ\} \leftrightarrow \forall x (φ \to A \subseteq x)$

Step	Hyp	Ref	Expression
1		ssint	$⊢ A \subseteq ⋂ \{x \| φ\} \leftrightarrow \forall y \in \{x \| φ\} A \subseteq y$
2		sseq2	$⊢ y = x \to (A \subseteq y \leftrightarrow A \subseteq x)$
3	2	ralab2	$⊢ \forall y \in \{x \| φ\} A \subseteq y \leftrightarrow \forall x (φ \to A \subseteq x)$
4	1 3	bitri	$⊢ A \subseteq ⋂ \{x \| φ\} \leftrightarrow \forall x (φ \to A \subseteq x)$