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Theorem opi2 3995
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 prexgOLD 3971 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
41, 2, 3mp2an 410 . . 3 {𝐴, 𝐵} ∈ V
54prid2 3502 . 2 {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
61, 2dfop 3573 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
75, 6eleqtrri 2127 1 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1407  Vcvv 2572  {csn 3400  {cpr 3401  cop 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-op 3409
This theorem is referenced by:  uniopel  4018  opeluu  4207  elvvuni  4429
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