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Theorem opi2 4216
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 4194 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
41, 2, 3mp2an 424 . . 3 {𝐴, 𝐵} ∈ V
54prid2 3688 . 2 {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}}
61, 2dfop 3762 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
75, 6eleqtrri 2246 1 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 2141  Vcvv 2730  {csn 3581  {cpr 3582  cop 3584
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590
This theorem is referenced by:  uniopel  4239  opeluu  4433  elvvuni  4673
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