Theorem bj-elpwgALT

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

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

Proof

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