Theorem elpw2g 3901

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)

Assertion

Ref	Expression
elpw2g	⊢ (B ∈ 𝑉 → (A ∈ 𝒫 B ↔ A ⊆ B))

Proof of Theorem elpw2g

Step	Hyp	Ref	Expression
1		elpwi 3360	. 2 ⊢ (A ∈ 𝒫 B → A ⊆ B)
2		ssexg 3887	. . . 4 ⊢ ((A ⊆ B ∧ B ∈ 𝑉) → A ∈ V)
3		elpwg 3359	. . . . 5 ⊢ (A ∈ V → (A ∈ 𝒫 B ↔ A ⊆ B))
4	3	biimparc 283	. . . 4 ⊢ ((A ⊆ B ∧ A ∈ V) → A ∈ 𝒫 B)
5	2, 4	syldan 266	. . 3 ⊢ ((A ⊆ B ∧ B ∈ 𝑉) → A ∈ 𝒫 B)
6	5	expcom 109	. 2 ⊢ (B ∈ 𝑉 → (A ⊆ B → A ∈ 𝒫 B))
7	1, 6	impbid2 131	1 ⊢ (B ∈ 𝑉 → (A ∈ 𝒫 B ↔ A ⊆ B))

Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  𝒫 cpw 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353

This theorem is referenced by:  elpw2  3902  pwnss  3903