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Theorem dfsn2 4138
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 4128 . 2 {𝐴, 𝐴} = ({𝐴} ∪ {𝐴})
2 unidm 3718 . 2 ({𝐴} ∪ {𝐴}) = {𝐴}
31, 2eqtr2i 2633 1 {𝐴} = {𝐴, 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  cun 3538  {csn 4125  {cpr 4127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-pr 4128
This theorem is referenced by:  nfsn  4189  disjprsn  4196  tpidm12  4234  tpidm  4237  preqsnd  4330  preqsn  4331  preqsnOLD  4332  elpreqprlem  4333  opid  4359  unisn  4387  intsng  4447  snex  4835  opeqsn  4892  propeqop  4895  relop  5194  funopg  5836  f1oprswap  6092  fnprb  6377  enpr1g  7908  supsn  8261  infsn  8293  prdom2  8712  wuntp  9412  wunsn  9417  grusn  9505  prunioo  12172  hashprg  13043  hashprgOLD  13044  hashfun  13084  lcmfsn  15186  lubsn  16917  indislem  20614  hmphindis  21410  wilthlem2  24595  upgrex  25759  umgrnloop0  25775  upgredg  25811  umgraex  25852  usgranloop0  25909  wlkntrllem1  26089  eupath2lem3  26506  1to2vfriswmgra  26533  esumpr2  29456  dvh2dim  35752  wopprc  36615  clsk1indlem4  37362  sge0prle  39294  meadjun  39355  opidg  40316  usgrnloop0ALT  40432  uspgr1v1eop  40475  1loopgruspgr  40715  1egrvtxdg0  40727  umgr2v2eedg  40740  umgr2v2e  40741  1wlkvtxeledglem  40827  ifpprsnss  40845  upgr1wlkwlk  40847  upgr11wlkdlem1  41312  1to2vfriswmgr  41449
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