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

Theorem elpwOLD 4506
 Description: Obsolete proof of elpw 4504 as of 31-Dec-2023. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 31-Dec-1993.)
Hypothesis
Ref Expression
elpwOLD.1 𝐴 ∈ V
Assertion
Ref Expression
elpwOLD (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Proof of Theorem elpwOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elpwOLD.1 . 2 𝐴 ∈ V
2 sseq1 3943 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 df-pw 4502 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
41, 2, 3elab2 3621 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2112  Vcvv 3444   ⊆ wss 3884  𝒫 cpw 4500 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator