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Theorem snidg 3450
Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
snidg (𝐴𝑉𝐴 ∈ {𝐴})

Proof of Theorem snidg
StepHypRef Expression
1 eqid 2085 . 2 𝐴 = 𝐴
2 elsng 3440 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
31, 2mpbiri 166 1 (𝐴𝑉𝐴 ∈ {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  {csn 3425
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-sn 3431
This theorem is referenced by:  snidb  3451  elsn2g  3454  snnzg  3534  snmg  3535  fvunsng  5436  fsnunfv  5442  1stconst  5924  2ndconst  5925  tfr0dm  6022  tfrlemibxssdm  6027  tfrlemi14d  6033  tfr1onlembxssdm  6043  tfr1onlemres  6049  tfrcllembxssdm  6056  tfrcllemres  6062  en1uniel  6454  onunsnss  6557  snon0  6573  supsnti  6621  fseq1p1m1  9415  elfzomin  9520  divalgmod  10721  bj-sels  11162
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