MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsn2 Structured version   Unicode version

Theorem dfsn2 3828
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 3821 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3490 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2457 1  |-  { A }  =  { A ,  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318   {csn 3814   {cpr 3815
This theorem is referenced by:  nfsn  3866  tpidm12  3905  tpidm  3908  preqsn  3980  opid  4002  unisn  4031  intsng  4085  snex  4405  opeqsn  4452  relop  5023  funopg  5485  f1oprswap  5717  enpr1g  7173  supsn  7474  prdom2  7890  wuntp  8586  wunsn  8591  grusn  8679  prunioo  11025  hashprg  11666  hashfun  11700  lubsn  14523  indislem  17064  hmphindis  17829  wilthlem2  20852  umgraex  21358  usgranloop0  21400  wlkntrllem1  21559  eupath2lem3  21701  preqsnd  24000  esumpr2  24458  wopprc  27101  1to2vfriswmgra  28396  dvh2dim  32243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-pr 3821
  Copyright terms: Public domain W3C validator