Theorem bj-elpwgALT 36391

Description: Alternate proof of elpwg 4597. See comment for bj-velpwALT 36390. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-elpwgALT	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Proof of Theorem bj-elpwgALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2813	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵))
2		sseq1 3999	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		bj-velpwALT 36390	. 2 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
4	1, 2, 3	vtoclbg 3537	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2098   ⊆ wss 3940  𝒫 cpw 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3947  df-ss 3957  df-pw 4596

This theorem is referenced by: (None)