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Theorem elopabran 5043
Description: Membership in a class abstraction of pairs, 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,𝐴,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem elopabran
StepHypRef Expression
1 simpl 472 . . . 4 ((𝑥𝑅𝑦𝜓) → 𝑥𝑅𝑦)
21ssopab2i 5032 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
32sseli 3632 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
4 elopabr 5042 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴𝑅)
53, 4syl 17 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} → 𝐴𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030   class class class wbr 4685  {copab 4745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746
This theorem is referenced by:  clwlkwlk  26727
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