Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfsn2 Structured version   GIF version

Theorem dfsn2 3381
 Description: Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
Assertion
Ref Expression
dfsn2 {A} = {A, A}

Proof of Theorem dfsn2
StepHypRef Expression
1 df-pr 3374 . 2 {A, A} = ({A} ∪ {A})
2 unidm 3080 . 2 ({A} ∪ {A}) = {A}
31, 2eqtr2i 2058 1 {A} = {A, A}
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∪ cun 2909  {csn 3367  {cpr 3368 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-v 2553  df-un 2916  df-pr 3374 This theorem is referenced by:  nfsn  3421  tpidm12  3460  tpidm  3463  preqsn  3537  opid  3558  unisn  3587  intsng  3640  opeqsn  3980  relop  4429  funopg  4877  enpr1g  6214  bj-snexg  9343
 Copyright terms: Public domain W3C validator