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Theorem dfsn2 3511
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 3504 . 2  |-  { A ,  A }  =  ( { A }  u.  { A } )
2 unidm 3189 . 2  |-  ( { A }  u.  { A } )  =  { A }
31, 2eqtr2i 2139 1  |-  { A }  =  { A ,  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1316    u. cun 3039   {csn 3497   {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-pr 3504
This theorem is referenced by:  nfsn  3553  tpidm12  3592  tpidm  3595  preqsn  3672  opid  3693  unisn  3722  intsng  3775  opeqsn  4144  relop  4659  funopg  5127  enpr1g  6660  hashprg  10522  bj-snexg  13037
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