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Theorem dfsn2 3788
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 3781 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3450 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2425 1  |-  { A }  =  { A ,  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3278   {csn 3774   {cpr 3775
This theorem is referenced by:  nfsn  3826  tpidm12  3865  tpidm  3868  preqsn  3940  opid  3962  unisn  3991  intsng  4045  snex  4365  opeqsn  4412  relop  4982  funopg  5444  f1oprswap  5676  enpr1g  7132  supsn  7430  prdom2  7846  wuntp  8542  wunsn  8547  grusn  8635  prunioo  10981  hashprg  11621  hashfun  11655  lubsn  14478  indislem  17019  hmphindis  17782  wilthlem2  20805  umgraex  21311  usgranloop0  21353  wlkntrllem1  21512  eupath2lem3  21654  preqsnd  23953  esumpr2  24411  wopprc  26991  1to2vfriswmgra  28110  dvh2dim  31928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-pr 3781
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