Theorem bj-velpwALT 37407

Description: This theorem bj-velpwALT 37407 and the next theorem bj-elpwgALT 37408 are alternate proofs of velpw 4541 and elpwg 4539 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3505 instead of proving first the general case using elab2g 3625 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2189. In other cases, that order is better (e.g., vsnex 5371 proved before snexg 5376). (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 4538	. . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
2	1	eleq2i 2832	. 2 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴})
3		abid 2722	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ 𝑥 ⊆ 𝐴)
4	2, 3	bitri 276	1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

Syntax hints:   ↔ wb 207   ∈ wcel 2119  {cab 2718   ⊆ wss 3890  𝒫 cpw 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-pw 4538

This theorem is referenced by:  bj-elpwgALT  37408