Theorem elpwOLD 4538

Description: Obsolete proof of elpw 4536 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 3985	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		df-pw 4534	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
4	1, 2, 3	elab2 3666	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 208   ∈ wcel 2113  Vcvv 3491   ⊆ wss 3929  𝒫 cpw 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-in 3936  df-ss 3945  df-pw 4534

This theorem is referenced by: (None)