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Theorem snid 3430
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 3429 . 2 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
31, 2mpbi 137 1 𝐴 ∈ {𝐴}
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sn 3409
This theorem is referenced by:  vsnid  3431  exsnrex  3441  rabsnt  3473  sneqr  3559  rext  3979  unipw  3981  intid  3988  snnex  4209  ordtriexmidlem2  4274  ordtriexmid  4275  ordtri2orexmid  4276  regexmidlem1  4286  0elsucexmid  4317  ordpwsucexmid  4322  opthprc  4419  fsn  5363  fsn2  5365  fvsn  5386  fvsnun1  5388  acexmidlema  5531  acexmidlemb  5532  acexmidlemab  5534  brtpos0  5898  en1  6310  elreal2  6965  1exp  9449  bj-d0clsepcl  10436
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