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Theorem elopabr 5565
Description: Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)
Assertion
Ref Expression
elopabr (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴𝑅)
Distinct variable groups:   𝑥,𝑅   𝑦,𝑅
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem elopabr
StepHypRef Expression
1 opabss 5214 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
21sseli 3976 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5150  {copab 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3473  df-in 3954  df-ss 3964  df-br 5151  df-opab 5213
This theorem is referenced by: (None)
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