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Theorem opi2 5460
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 5423 . . 3 {𝐴, 𝐵} ∈ V
21prid2 4760 . 2 {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
3 opi1.1 . . 3 𝐴 ∈ V
4 opi1.2 . . 3 𝐵 ∈ V
53, 4dfop 4865 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
62, 5eleqtrri 2824 1 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3466  {csn 4621  {cpr 4623  cop 4627
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 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628
This theorem is referenced by:  opeluu  5461  uniopel  5507  elvvuni  5743
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