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)

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

Proof

Step	Hyp	Ref	Expression
1		exsimpl	$$⊢\exists x\left(x=A\wedge x\in B\right)\to \exists xx=A$$
2		dfclel	$$⊢A\in B\leftrightarrow \exists x\left(x=A\wedge x\in B\right)$$
3		isset	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
4	1 2 3	3imtr4i	$$⊢A\in B\to A\in \mathrm{V}$$