Theorem bj-velpwALT 37066

Description: This theorem bj-velpwALT 37066 and the next theorem bj-elpwgALT 37067 are alternate proofs of velpw 4553 and elpwg 4551 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3510 instead of proving first the general case using elab2g 3634 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2179. In other cases, that order is better (e.g., vsnex 5370 proved before snexg 5371). (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-velpwALT	⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-velpwALT

Step	Hyp	Ref	Expression
1		df-pw 4550	. . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
2	1	eleq2i 2821	. 2 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴})
3		abid 2712	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ 𝑥 ⊆ 𝐴)
4	2, 3	bitri 275	1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2110  {cab 2708   ⊆ wss 3900  𝒫 cpw 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-12 2179  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-pw 4550

This theorem is referenced by:  bj-elpwgALT  37067