Theorem elpwgdedVD

Description: Membership in a power class. Theorem 86 of Suppes p. 47. Derived from elpwg . In form of VD deduction with ph and ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded is elpwgdedVD using conventional notation. (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	elpwgdedVD.1	$$⊢\left(\phi \to A\in \mathrm{V}\right)$$
	elpwgdedVD.2	$$⊢\left(\psi \to A\subseteq B\right)$$
Assertion	elpwgdedVD	$$⊢\left(\left(\phi ,\psi \right)\to A\in 𝒫B\right)$$

Proof

Step	Hyp	Ref	Expression
1		elpwgdedVD.1	$$⊢\left(\phi \to A\in \mathrm{V}\right)$$
2		elpwgdedVD.2	$$⊢\left(\psi \to A\subseteq B\right)$$
3		elpwg	$$⊢A\in \mathrm{V}\to \left(A\in 𝒫B\leftrightarrow A\subseteq B\right)$$
4	3	biimpar	$$⊢\left(A\in \mathrm{V}\wedge A\subseteq B\right)\to A\in 𝒫B$$
5	1 2 4	el12	$$⊢\left(\left(\phi ,\psi \right)\to A\in 𝒫B\right)$$