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

Theorem elopabw 5481
Description: Membership in a class abstraction of ordered pairs. Weaker version of elopab 5482 with a sethood antecedent, avoiding ax-sep 5254, ax-nul 5261, and ax-pr 5382. Originally a subproof of elopab 5482. (Contributed by SN, 11-Dec-2024.)
Assertion
Ref Expression
elopabw (𝐴𝑉 → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elopabw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2741 . . . 4 (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
21anbi1d 630 . . 3 (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
322exbidv 1927 . 2 (𝑧 = 𝐴 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4 df-opab 5166 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
53, 4elab2g 3630 1 (𝐴𝑉 → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  cop 4590  {copab 5165
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-opab 5166
This theorem is referenced by:  elopab  5482  iunopab  5514  ssrel  5736
  Copyright terms: Public domain W3C validator