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Theorem opi1 5479
Description: One of the two elements in an ordered pair. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
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 snex 5442 . . 3 {𝐴} ∈ V
21prid1 4767 . 2 {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
3 opi1.1 . . 3 𝐴 ∈ V
4 opi1.2 . . 3 𝐵 ∈ V
53, 4dfop 4877 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
62, 5eleqtrri 2838 1 {𝐴} ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  {csn 4631  {cpr 4633  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  opth1  5486  opth  5487
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