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Theorem elopab 4243
Description: Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elopab (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem elopab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 2741 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
2 vex 2733 . . . . . 6 𝑥 ∈ V
3 vex 2733 . . . . . 6 𝑦 ∈ V
42, 3opex 4214 . . . . 5 𝑥, 𝑦⟩ ∈ V
5 eleq1 2233 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V))
64, 5mpbiri 167 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
76adantr 274 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
87exlimivv 1889 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
9 eqeq1 2177 . . . . 5 (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
109anbi1d 462 . . . 4 (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
11102exbidv 1861 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
12 df-opab 4051 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
1311, 12elab2g 2877 . 2 (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
141, 8, 13pm5.21nii 699 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  Vcvv 2730  cop 3586  {copab 4049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051
This theorem is referenced by:  opelopabsbALT  4244  opelopabsb  4245  opelopabt  4247  opelopabga  4248  opabm  4265  iunopab  4266  epelg  4275  elxp  4628  elco  4777  elcnv  4788  dfmpt3  5320  0neqopab  5898  brabvv  5899  opabex3d  6100  opabex3  6101
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