MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elpwgOLD Structured version   Visualization version   GIF version

Theorem elpwgOLD 4504
Description: Obsolete proof of elpwg 4500 as of 31-Dec-2023. (Contributed by NM, 6-Aug-2000.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elpwgOLD (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Proof of Theorem elpwgOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2877 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3940 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 velpw 4502 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
41, 2, 3vtoclbg 3517 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2111  wss 3881  𝒫 cpw 4497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator