class2set

Description: The class of elements of A "such that A is a set" is a set. That class is equal to A when A is a set (see class2seteq ) and to the empty set when A is a proper class. (Contributed by NM, 16-Oct-2003)

		Ref	Expression
	Assertion	class2set	$⊢ \{x \in A \| A \in V\} \in V$

Proof

Step	Hyp	Ref	Expression
1		rabexg	$⊢ A \in V \to \{x \in A \| A \in V\} \in V$
2		simpl	$⊢ (\neg A \in V \land x \in A) \to \neg A \in V$
3	2	nrexdv	$⊢ \neg A \in V \to \neg \exists x \in A A \in V$
4		rabn0	$⊢ \{x \in A \| A \in V\} \neq \emptyset \leftrightarrow \exists x \in A A \in V$
5	4	necon1bbii	$⊢ \neg \exists x \in A A \in V \leftrightarrow \{x \in A \| A \in V\} = \emptyset$
6	3 5	sylib	$⊢ \neg A \in V \to \{x \in A \| A \in V\} = \emptyset$
7		0ex	$⊢ \emptyset \in V$
8	6 7	eqeltrdi	$⊢ \neg A \in V \to \{x \in A \| A \in V\} \in V$
9	1 8	pm2.61i	$⊢ \{x \in A \| A \in V\} \in V$

Metamath Proof Explorer

Theorem class2set

Proof