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Theorem opi1 4908
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 4879 . . 3 {𝐴} ∈ V
21prid1 4274 . 2 {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}}
3 opi1.1 . . 3 𝐴 ∈ V
4 opi1.2 . . 3 𝐵 ∈ V
53, 4dfop 4376 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
62, 5eleqtrri 2697 1 {𝐴} ∈ ⟨𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  Vcvv 3190  {csn 4155  {cpr 4157  cop 4161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162
This theorem is referenced by:  opth1  4914  opth  4915
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