ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opi2 Structured version   GIF version

Theorem opi2 3961
Description: One of the two elements of 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
opi2 {A, B} A, B

Proof of Theorem opi2
StepHypRef Expression
1 opi1.1 . . . 4 A V
2 opi1.2 . . . 4 B V
3 prexgOLD 3937 . . . 4 ((A V B V) → {A, B} V)
41, 2, 3mp2an 402 . . 3 {A, B} V
54prid2 3468 . 2 {A, B} {{A}, {A, B}}
61, 2dfop 3539 . 2 A, B⟩ = {{A}, {A, B}}
75, 6eleqtrri 2110 1 {A, B} A, B
Colors of variables: wff set class
Syntax hints:   wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368  cop 3370
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  uniopel  3984  opeluu  4148  elvvuni  4347
  Copyright terms: Public domain W3C validator