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Theorem opi1 5337
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 5309 . . 3 {𝐴} ∈ V
21prid1 4672 . 2 {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
3 opi1.1 . . 3 𝐴 ∈ V
4 opi1.2 . . 3 𝐵 ∈ V
53, 4dfop 4775 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
62, 5eleqtrri 2913 1 {𝐴} ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2114  Vcvv 3469  {csn 4539  {cpr 4541  cop 4545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546
This theorem is referenced by:  opth1  5344  opth  5345
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