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Theorem bj-elpwgALT 36391
Description: Alternate proof of elpwg 4597. See comment for bj-velpwALT 36390. (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 2813 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3999 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 bj-velpwALT 36390 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
41, 2, 3vtoclbg 3537 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wss 3940  𝒫 cpw 4594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3947  df-ss 3957  df-pw 4596
This theorem is referenced by: (None)
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