MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elopabran Structured version   Visualization version   GIF version

Theorem elopabran 5571
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 5559 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
3 opabss 5211 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
42, 3sstri 4004 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ⊆ 𝑅
54sseli 3990 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} → 𝐴𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105   class class class wbr 5147  {copab 5209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ss 3979  df-br 5148  df-opab 5210
This theorem is referenced by:  opabresex2  7484  fvmptopab  7486  clwlkwlk  29807
  Copyright terms: Public domain W3C validator