Theorem bj-velpwALT 37036

Description: This theorem bj-velpwALT 37036 and the next theorem bj-elpwgALT 37037 are alternate proofs of velpw 4570 and elpwg 4568 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3526 instead of proving first the general case using elab2g 3649 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2178. In other cases, that order is better (e.g., vsnex 5391 proved before snexg 5392). (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 4567	. . 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 2109  {cab 2708   ⊆ wss 3916  𝒫 cpw 4565

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-pw 4567

This theorem is referenced by:  bj-elpwgALT  37037