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Theorem opi1 3969
 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
2 snexgOLD 3935 . . . 4 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 7 . . 3 {𝐴} ∈ V
43prid1 3476 . 2 {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
5 opi1.2 . . 3 𝐵 ∈ V
61, 5dfop 3548 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
74, 6eleqtrri 2113 1 {𝐴} ∈ ⟨𝐴, 𝐵
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  Vcvv 2557  {csn 3375  {cpr 3376  ⟨cop 3378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384 This theorem is referenced by:  opth1  3973  opth  3974
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