Theorem elpwgOLD 4506

Description: Obsolete proof of elpwg 4502 as of 31-Dec-2023. (Contributed by NM, 6-Aug-2000.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem elpwgOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2818	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵))
2		sseq1 3912	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		velpw 4504	. 2 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
4	1, 2, 3	vtoclbg 3473	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2112   ⊆ wss 3853  𝒫 cpw 4499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-in 3860  df-ss 3870  df-pw 4501

This theorem is referenced by: (None)