Theorem elpwi2OLD 5210

Description: Obsolete version of elpwi2 5209 as of 26-May-2024. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elpwi2.1	⊢ 𝐵 ∈ 𝑉
elpwi2.2	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
elpwi2OLD	⊢ 𝐴 ∈ 𝒫 𝐵

Proof of Theorem elpwi2OLD

Step	Hyp	Ref	Expression
1		elpwi2.2	. 2 ⊢ 𝐴 ⊆ 𝐵
2		elpwi2.1	. . 3 ⊢ 𝐵 ∈ 𝑉
3		elpw2g 5207	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	2, 3	ax-mp 5	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 234	1 ⊢ 𝐴 ∈ 𝒫 𝐵

Syntax hints:   ↔ wb 209   ∈ wcel 2112   ⊆ wss 3854  𝒫 cpw 4487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730  ax-sep 5162

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3077  df-v 3409  df-in 3861  df-ss 3871  df-pw 4489

This theorem is referenced by: (None)