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

Theorem opprc 3561
Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc  _V  _V  <. ,  >.  (/)

Proof of Theorem opprc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-op 3376 . 2  <. ,  >.  {  |  _V  _V  { { } ,  { ,  } } }
2 3simpa 900 . . . . 5  _V  _V  { { } ,  { ,  } }  _V  _V
32con3i 561 . . . 4  _V  _V  _V  _V  { { } ,  { ,  } }
43alrimiv 1751 . . 3  _V  _V  _V  _V 
{ { } ,  { ,  } }
5 abeq0 3242 . . 3  {  |  _V  _V  { { } ,  { ,  } } }  (/)  _V  _V 
{ { } ,  { ,  } }
64, 5sylibr 137 . 2  _V  _V  {  |  _V  _V  { { } ,  { ,  } } }  (/)
71, 6syl5eq 2081 1  _V  _V  <. ,  >.  (/)
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   w3a 884  wal 1240   wceq 1242   wcel 1390   {cab 2023   _Vcvv 2551   (/)c0 3218   {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-in1 544  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-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-op 3376
This theorem is referenced by:  opprc1  3562  opprc2  3563  ovprc  5482
  Copyright terms: Public domain W3C validator