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Theorem dfsn2 4040
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 4030 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3647 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2497 1  |-  { A }  =  { A ,  A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    u. cun 3474   {csn 4027   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-pr 4030
This theorem is referenced by:  nfsn  4085  tpidm12  4128  tpidm  4131  preqsn  4209  opid  4232  unisn  4260  intsng  4317  snex  4688  opeqsn  4743  relop  5151  funopg  5618  f1oprswap  5853  fnprb  6117  enpr1g  7578  supsn  7926  prdom2  8380  wuntp  9085  wunsn  9090  grusn  9178  prunioo  11645  hashprg  12424  hashfun  12457  lubsn  15577  indislem  19267  hmphindis  20033  wilthlem2  23071  umgraex  23999  usgranloop0  24056  wlkntrllem1  24237  eupath2lem3  24655  1to2vfriswmgra  24682  preqsnd  27092  esumpr2  27714  wopprc  30576  dvh2dim  36242
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