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Theorem opi2 5427
Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Hypotheses
Ref Expression
opi1.1 𝐴 ∈ V
opi1.2 𝐵 ∈ V
Assertion
Ref Expression
opi2 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵

Proof of Theorem opi2
StepHypRef Expression
1 prex 5390 . . 3 {𝐴, 𝐵} ∈ V
21prid2 4725 . 2 {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
3 opi1.1 . . 3 𝐴 ∈ V
4 opi1.2 . . 3 𝐵 ∈ V
53, 4dfop 4830 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
62, 5eleqtrri 2833 1 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3444  {csn 4587  {cpr 4589  cop 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594
This theorem is referenced by:  opeluu  5428  uniopel  5474  elvvuni  5709
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