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	bm1.3ii	$$⊢\exists y\forall z\left(z\in y\leftrightarrow z\subseteq x\right)$$
3		bicom	$$⊢\left(z\subseteq x\leftrightarrow z\in y\right)\leftrightarrow \left(z\in y\leftrightarrow z\subseteq x\right)$$
4	3	albii	$$⊢\forall z\left(z\subseteq x\leftrightarrow z\in y\right)\leftrightarrow \forall z\left(z\in y\leftrightarrow z\subseteq x\right)$$
5	4	exbii	$$⊢\exists y\forall z\left(z\subseteq x\leftrightarrow z\in y\right)\leftrightarrow \exists y\forall z\left(z\in y\leftrightarrow z\subseteq x\right)$$
6	2 5	mpbir	$$⊢\exists y\forall z\left(z\subseteq x\leftrightarrow z\in y\right)$$