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Theorem dfsn2 3730
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 3723 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3394 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2379 1  |-  { A }  =  { A ,  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1642    u. cun 3226   {csn 3716   {cpr 3717
This theorem is referenced by:  nfsn  3767  tpidm12  3804  tpidm  3807  preqsn  3873  opid  3895  unisn  3924  intsng  3978  snex  4297  opeqsn  4344  relop  4916  funopg  5368  f1oprswap  5598  enpr1g  7015  supsn  7310  prdom2  7726  wuntp  8423  wunsn  8428  grusn  8516  prunioo  10856  hashprg  11464  hashfun  11485  lubsn  14299  indislem  16843  hmphindis  17594  wilthlem2  20419  preqsnd  23199  esumpr2  23724  umgraex  24279  eupath2lem3  24307  wopprc  26446  usgranloop0  27554  wlkntrllem1  27701  1to2vfriswmgra  27839  dvh2dim  31704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-un 3233  df-pr 3723
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