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Theorem opi1 4059
Description: One of the two elements in 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
opi1 {𝐴} ∈ ⟨𝐴, 𝐵

Proof of Theorem opi1
StepHypRef Expression
1 opi1.1 . . . 4 𝐴 ∈ V
21snex 4020 . . 3 {𝐴} ∈ V
32prid1 3548 . 2 {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
4 opi1.2 . . 3 𝐵 ∈ V
51, 4dfop 3621 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
63, 5eleqtrri 2163 1 {𝐴} ∈ ⟨𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1438  Vcvv 2619  {csn 3446  {cpr 3447  cop 3449
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455
This theorem is referenced by:  opth1  4063  opth  4064
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