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Theorem opi1 4154
 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 4109 . . 3 {𝐴} ∈ V
32prid1 3629 . 2 {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
4 opi1.2 . . 3 𝐵 ∈ V
51, 4dfop 3704 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
63, 5eleqtrri 2215 1 {𝐴} ∈ ⟨𝐴, 𝐵
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1480  Vcvv 2686  {csn 3527  {cpr 3528  ⟨cop 3530 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536 This theorem is referenced by:  opth1  4158  opth  4159
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