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

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

Proof of Theorem iunid
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-sn 3373 . . . . 5 {x} = {yy = x}
2 equcom 1590 . . . . . 6 (y = xx = y)
32abbii 2150 . . . . 5 {yy = x} = {yx = y}
41, 3eqtri 2057 . . . 4 {x} = {yx = y}
54a1i 9 . . 3 (x A → {x} = {yx = y})
65iuneq2i 3666 . 2 x A {x} = x A {yx = y}
7 iunab 3694 . . 3 x A {yx = y} = {yx A x = y}
8 risset 2346 . . . 4 (y Ax A x = y)
98abbii 2150 . . 3 {yy A} = {yx A x = y}
10 abid2 2155 . . 3 {yy A} = A
117, 9, 103eqtr2i 2063 . 2 x A {yx = y} = A
126, 11eqtri 2057 1 x A {x} = A
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  {csn 3367  ∪ 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-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-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