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Theorem dfsn2 3984
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 3974 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3585 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2432 1  |-  { A }  =  { A ,  A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    u. cun 3411   {csn 3971   {cpr 3973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-un 3418  df-pr 3974
This theorem is referenced by:  nfsn  4028  tpidm12  4072  tpidm  4075  preqsn  4154  opid  4177  unisn  4205  intsng  4262  snex  4631  opeqsn  4685  relop  4973  funopg  5600  f1oprswap  5837  fnprb  6109  enpr1g  7618  supsn  7963  prdom2  8415  wuntp  9118  wunsn  9123  grusn  9211  prunioo  11701  hashprg  12507  hashfun  12542  lubsn  16046  indislem  19791  hmphindis  20588  wilthlem2  23722  umgraex  24727  usgranloop0  24784  wlkntrllem1  24965  eupath2lem3  25383  1to2vfriswmgra  25410  preqsnd  27825  esumpr2  28500  dvh2dim  34445  wopprc  35314
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