ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfsn2 GIF version

Theorem dfsn2 3389
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 3382 . 2 {𝐴, 𝐴} = ({𝐴} ∪ {𝐴})
2 unidm 3086 . 2 ({𝐴} ∪ {𝐴}) = {𝐴}
31, 2eqtr2i 2061 1 {𝐴} = {𝐴, 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1243  cun 2915  {csn 3375  {cpr 3376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-pr 3382
This theorem is referenced by:  nfsn  3430  tpidm12  3469  tpidm  3472  preqsn  3546  opid  3567  unisn  3596  intsng  3649  opeqsn  3989  relop  4486  funopg  4934  enpr1g  6278  bj-snexg  10006
  Copyright terms: Public domain W3C validator