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Theorem bj-elpwgALT 37055
Description: Alternate proof of elpwg 4603. See comment for bj-velpwALT 37054. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-elpwgALT (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Proof of Theorem bj-elpwgALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2829 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 4009 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 bj-velpwALT 37054 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
41, 2, 3vtoclbg 3557 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wss 3951  𝒫 cpw 4600
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 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ss 3968  df-pw 4602
This theorem is referenced by: (None)
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