Theorem elex

Description: If a class is a member of another class, then it is a set. Theorem 6.12 of Quine p. 44. (Contributed by NM, 26-May-1993) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof shortened by Wolf Lammen, 28-May-2025)

	Ref	Expression
Assertion	elex	$$⊢A\in B\to A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		elissetv	$$⊢A\in B\to \exists xx=A$$
2		isset	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
3	1 2	sylibr	$$⊢A\in B\to A\in \mathrm{V}$$