Theorem bj-velpwALT 36468

Description: This theorem bj-velpwALT 36468 and the next theorem bj-elpwgALT 36469 are alternate proofs of velpw 4603 and elpwg 4601 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3540 instead of proving first the general case using elab2g 3667 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2164. In other cases, that order is better (e.g., vsnex 5425 proved before snexg 5426). (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 4600	. . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
2	1	eleq2i 2820	. 2 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴})
3		abid 2708	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ 𝑥 ⊆ 𝐴)
4	2, 3	bitri 275	1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2099  {cab 2704   ⊆ wss 3944  𝒫 cpw 4598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-pw 4600

This theorem is referenced by:  bj-elpwgALT  36469