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Theorem elopabran 5555
Description: Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.)
Assertion
Ref Expression
elopabran (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} → 𝐴𝑅)
Distinct variable groups:   𝑥,𝑅   𝑦,𝑅
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem elopabran
StepHypRef Expression
1 simpl 482 . . . 4 ((𝑥𝑅𝑦𝜓) → 𝑥𝑅𝑦)
21ssopab2i 5543 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
3 opabss 5205 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
42, 3sstri 3986 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ⊆ 𝑅
54sseli 3973 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} → 𝐴𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098   class class class wbr 5141  {copab 5203
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 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-br 5142  df-opab 5204
This theorem is referenced by:  opabresex2  7457  fvmptopab  7459  clwlkwlk  29541
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