# Metamath Proof Explorer

## 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 ${⊢}\left(\left({\phi },{\psi }\right){\to }{A}\in 𝒫{B}\right)$

### 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 ${⊢}{A}\in \mathrm{V}\to \left({A}\in 𝒫{B}↔{A}\subseteq {B}\right)$
4 3 biimpar ${⊢}\left({A}\in \mathrm{V}\wedge {A}\subseteq {B}\right)\to {A}\in 𝒫{B}$
5 1 2 4 el12 ${⊢}\left(\left({\phi },{\psi }\right){\to }{A}\in 𝒫{B}\right)$