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

Theorem iunid 3703
Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
Assertion
Ref Expression
iunid  U_  { }
Distinct variable group:   ,

Proof of Theorem iunid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-sn 3373 . . . . 5  { }  {  |  }
2 equcom 1590 . . . . . 6
32abbii 2150 . . . . 5  {  |  }  {  |  }
41, 3eqtri 2057 . . . 4  { }  {  |  }
54a1i 9 . . 3  { }  {  |  }
65iuneq2i 3666 . 2  U_  { }  U_  {  |  }
7 iunab 3694 . . 3  U_  {  |  }  {  |  }
8 risset 2346 . . . 4
98abbii 2150 . . 3  {  |  }  {  |  }
10 abid2 2155 . . 3  {  |  }
117, 9, 103eqtr2i 2063 . 2  U_  {  |  }
126, 11eqtri 2057 1  U_  { }
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390   {cab 2023  wrex 2301   {csn 3367   U_ciun 3648
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-sn 3373  df-iun 3650
This theorem is referenced by:  iunxpconst  4343  xpexgALT  5702  uniqs  6100
  Copyright terms: Public domain W3C validator