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

Theorem opthpr 3534
Description: A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
Hypotheses
Ref Expression
preq12b.1  _V
preq12b.2  _V
preq12b.3  C 
_V
preq12b.4  D 
_V
Assertion
Ref Expression
opthpr  =/=  D  { ,  }  { C ,  D }  C  D

Proof of Theorem opthpr
StepHypRef Expression
1 preq12b.1 . . 3  _V
2 preq12b.2 . . 3  _V
3 preq12b.3 . . 3  C 
_V
4 preq12b.4 . . 3  D 
_V
51, 2, 3, 4preq12b 3532 . 2  { ,  }  { C ,  D }  C  D  D  C
6 idd 21 . . . 4  =/=  D  C  D  C  D
7 df-ne 2203 . . . . . 6  =/=  D  D
8 pm2.21 547 . . . . . 6  D  D  C  C  D
97, 8sylbi 114 . . . . 5  =/=  D  D  C  C  D
109impd 242 . . . 4  =/=  D  D  C  C  D
116, 10jaod 636 . . 3  =/=  D  C  D  D  C  C  D
12 orc 632 . . 3  C  D  C  D  D  C
1311, 12impbid1 130 . 2  =/=  D  C  D  D  C  C  D
145, 13syl5bb 181 1  =/=  D  { ,  }  { C ,  D }  C  D
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   wceq 1242   wcel 1390    =/= wne 2201   _Vcvv 2551   {cpr 3368
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-in2 545  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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator