Theorem elpwOLD 4505

Description: Obsolete proof of elpw 4503 as of 31-Dec-2023. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 31-Dec-1993.)

Hypothesis

Ref	Expression
elpwOLD.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elpwOLD	⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)

Proof of Theorem elpwOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwOLD.1	. 2 ⊢ 𝐴 ∈ V
2		sseq1 3912	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		df-pw 4501	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
4	1, 2, 3	elab2 3580	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 209   ∈ wcel 2112  Vcvv 3398   ⊆ 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)