MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsn2 Structured version   Unicode version

Theorem dfsn2 3885
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 3875 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3494 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2459 1  |-  { A }  =  { A ,  A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    u. cun 3321   {csn 3872   {cpr 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969  df-un 3328  df-pr 3875
This theorem is referenced by:  nfsn  3929  tpidm12  3971  tpidm  3974  preqsn  4050  opid  4073  unisn  4101  intsng  4158  snex  4528  opeqsn  4582  relop  4985  funopg  5445  f1oprswap  5675  fnprb  5931  enpr1g  7367  supsn  7711  prdom2  8165  wuntp  8870  wunsn  8875  grusn  8963  prunioo  11406  hashprg  12147  hashfun  12191  lubsn  15256  indislem  18579  hmphindis  19345  wilthlem2  22382  umgraex  23208  usgranloop0  23250  wlkntrllem1  23409  eupath2lem3  23551  preqsnd  25852  esumpr2  26469  wopprc  29332  1to2vfriswmgra  30551  dvh2dim  34930
  Copyright terms: Public domain W3C validator