Theorem axpow3

Description: A variant of the Axiom of Power Sets ax-pow . For any set x , there exists a set y whose members are exactly the subsets of x i.e. the power set of x . Axiom Pow of BellMachover p. 466. (Contributed by NM, 4-Jun-2006)

	Ref	Expression
Assertion	axpow3	$$⊢\exists y\forall z\left(z\subseteq x\leftrightarrow z\in y\right)$$

Proof

Step	Hyp	Ref	Expression
1		axpow2	$$⊢\exists y\forall z\left(z\subseteq x\to z\in y\right)$$
2	1	sepexi	$$⊢\exists y\forall z\left(z\in y\leftrightarrow z\subseteq x\right)$$
3		bicom1	$$⊢\left(z\in y\leftrightarrow z\subseteq x\right)\to \left(z\subseteq x\leftrightarrow z\in y\right)$$
4	3	alimi	$$⊢\forall z\left(z\in y\leftrightarrow z\subseteq x\right)\to \forall z\left(z\subseteq x\leftrightarrow z\in y\right)$$
5	2 4	eximii	$$⊢\exists y\forall z\left(z\subseteq x\leftrightarrow z\in y\right)$$