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Theorem dfsn2 3509
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 3502 . 2 {𝐴, 𝐴} = ({𝐴} ∪ {𝐴})
2 unidm 3187 . 2 ({𝐴} ∪ {𝐴}) = {𝐴}
31, 2eqtr2i 2137 1 {𝐴} = {𝐴, 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1314  cun 3037  {csn 3495  {cpr 3496
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-pr 3502
This theorem is referenced by:  nfsn  3551  tpidm12  3590  tpidm  3593  preqsn  3670  opid  3691  unisn  3720  intsng  3773  opeqsn  4142  relop  4657  funopg  5125  enpr1g  6658  hashprg  10494  bj-snexg  12912
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