Theorem axpow2

Description: A variant of the Axiom of Power Sets ax-pow using subset notation. Problem in BellMachover p. 466. (Contributed by NM, 4-Jun-2006)

	Ref	Expression
Assertion	axpow2	$$⊢\exists y\forall z\left(z\subseteq x\to z\in y\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-pow	$$⊢\exists y\forall z\left(\forall w\left(w\in z\to w\in x\right)\to z\in y\right)$$
2		dfss2	$$⊢z\subseteq x\leftrightarrow \forall w\left(w\in z\to w\in x\right)$$
3	2	imbi1i	$$⊢\left(z\subseteq x\to z\in y\right)\leftrightarrow \left(\forall w\left(w\in z\to w\in x\right)\to z\in y\right)$$
4	3	albii	$$⊢\forall z\left(z\subseteq x\to z\in y\right)\leftrightarrow \forall z\left(\forall w\left(w\in z\to w\in x\right)\to z\in y\right)$$
5	4	exbii	$$⊢\exists y\forall z\left(z\subseteq x\to z\in y\right)\leftrightarrow \exists y\forall z\left(\forall w\left(w\in z\to w\in x\right)\to z\in y\right)$$
6	1 5	mpbir	$$⊢\exists y\forall z\left(z\subseteq x\to z\in y\right)$$