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 e. B -> A e. _V )

Proof

Step	Hyp	Ref	Expression
1		elissetv	|- ( A e. B -> E. x x = A )
2		isset	|- ( A e. _V <-> E. x x = A )
3	1 2	sylibr	|- ( A e. B -> A e. _V )