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Theorem opi2 4125
Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
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 opi1.1 . . . 4 𝐴 ∈ V
2 opi1.2 . . . 4 𝐵 ∈ V
3 prexg 4103 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
41, 2, 3mp2an 422 . . 3 {𝐴, 𝐵} ∈ V
54prid2 3600 . 2 {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
61, 2dfop 3674 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
75, 6eleqtrri 2193 1 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1465  Vcvv 2660  {csn 3497  {cpr 3498  cop 3500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506
This theorem is referenced by:  uniopel  4148  opeluu  4341  elvvuni  4573
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