Theorem bj-velpwALT 38043

Description: This theorem bj-velpwALT 38043 and the next theorem bj-elpwgALT 38044 are alternate proofs of velpw 4561 and elpwg 4559 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3519 instead of proving first the general case using elab2g 3633 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2212. In other cases, that order is better (e.g., vsnex 5392 proved before snexg 5397). (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 4558	. . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
2	1	eleq2i 2852	. 2 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴})
3		abid 2742	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ 𝑥 ⊆ 𝐴)
4	2, 3	bitri 278	1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2144  {cab 2738   ⊆ wss 3895  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-pw 4558

This theorem is used by:  bj-elpwgALT  38044