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Theorem snid 3475
Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
Hypothesis
Ref Expression
snid.1 𝐴 ∈ V
Assertion
Ref Expression
snid 𝐴 ∈ {𝐴}

Proof of Theorem snid
StepHypRef Expression
1 snid.1 . 2 𝐴 ∈ V
2 snidb 3474 . 2 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
31, 2mpbi 143 1 𝐴 ∈ {𝐴}
Colors of variables: wff set class
Syntax hints:  wcel 1438  Vcvv 2619  {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sn 3452
This theorem is referenced by:  vsnid  3476  exsnrex  3485  rabsnt  3517  sneqr  3604  undifexmid  4028  exmidexmid  4031  exmid01  4032  exmidundif  4035  unipw  4044  intid  4051  ordtriexmidlem2  4337  ordtriexmid  4338  ordtri2orexmid  4339  regexmidlem1  4349  0elsucexmid  4381  ordpwsucexmid  4386  opthprc  4489  fsn  5469  fsn2  5471  fvsn  5492  fvsnun1  5494  acexmidlema  5643  acexmidlemb  5644  acexmidlemab  5646  brtpos0  6017  mapsn  6447  mapsncnv  6452  en1  6516  djulclr  6741  djurclr  6742  djulcl  6743  djurcl  6744  djuf1olem  6745  elreal2  7368  1exp  9984  hashinfuni  10185  djucllem  11700  bj-d0clsepcl  11820
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