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Theorem elopabr 5415
Description: Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.)
Assertion
Ref Expression
elopabr (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴𝑅)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem elopabr
StepHypRef Expression
1 elopab 5382 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
2 df-br 5031 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
32biimpi 219 . . . . 5 (𝑥𝑅𝑦 → ⟨𝑥, 𝑦⟩ ∈ 𝑅)
4 eleq1 2820 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
53, 4syl5ibr 249 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝑅𝑦𝐴𝑅))
65imp 410 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴𝑅)
76exlimivv 1939 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴𝑅)
81, 7sylbi 220 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wex 1786  wcel 2114  cop 4522   class class class wbr 5030  {copab 5092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093
This theorem is referenced by:  elopabran  5416
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