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

Theorem opabss 3812
Description: The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
opabss  { <. ,  >.  |  R }  C_  R
Distinct variable groups:   , R   , R

Proof of Theorem opabss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-opab 3810 . 2  { <. ,  >.  |  R }  {  |  <. , 
>.  R }
2 df-br 3756 . . . . 5  R  <. ,  >.  R
3 eleq1 2097 . . . . . 6  <. , 
>.  R  <. , 
>.  R
43biimpar 281 . . . . 5  <. ,  >.  <. ,  >.  R  R
52, 4sylan2b 271 . . . 4  <. ,  >.  R  R
65exlimivv 1773 . . 3  <. , 
>.  R  R
76abssi 3009 . 2  {  |  <. ,  >.  R }  C_  R
81, 7eqsstri 2969 1  { <. ,  >.  |  R }  C_  R
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242  wex 1378   wcel 1390   {cab 2023    C_ wss 2911   <.cop 3370   class class class wbr 3755   {copab 3808
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-bndl 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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator