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Theorem opi2 4024
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 4002 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
41, 2, 3mp2an 417 . . 3 {𝐴, 𝐵} ∈ V
54prid2 3523 . 2 {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
61, 2dfop 3595 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
75, 6eleqtrri 2158 1 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1434  Vcvv 2612  {csn 3422  {cpr 3423  cop 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431
This theorem is referenced by:  uniopel  4047  opeluu  4236  elvvuni  4460
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