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

Theorem eusn 3435
 Description: Two ways to express "A is a singleton." (Contributed by NM, 30-Oct-2010.)
Assertion
Ref Expression
eusn (∃!x x Ax A = {x})
Distinct variable group:   x,A

Proof of Theorem eusn
StepHypRef Expression
1 euabsn 3431 . 2 (∃!x x Ax{xx A} = {x})
2 abid2 2155 . . . 4 {xx A} = A
32eqeq1i 2044 . . 3 ({xx A} = {x} ↔ A = {x})
43exbii 1493 . 2 (x{xx A} = {x} ↔ x A = {x})
51, 4bitri 173 1 (∃!x x Ax A = {x})
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  {cab 2023  {csn 3367 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-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-sn 3373 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator