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Theorem dfsn2 3654
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 3647 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3318 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2304 1  |-  { A }  =  { A ,  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150   {csn 3640   {cpr 3641
This theorem is referenced by:  nfsn  3691  tpidm12  3728  tpidm  3731  preqsn  3792  opid  3814  unisn  3843  intsng  3897  snex  4216  opeqsn  4262  relop  4834  funopg  5286  f1oprswap  5515  enpr1g  6927  supsn  7220  prdom2  7636  wuntp  8333  wunsn  8338  grusn  8426  prunioo  10764  hashprg  11368  hashfun  11389  lubsn  14200  indislem  16737  hmphindis  17488  wilthlem2  20307  preqsnd  23194  esumpr2  23439  umgraex  23875  eupath2lem3  23903  singempcon  25593  wopprc  27123  1to2vfriswmgra  28184  dvh2dim  31635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-pr 3647
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