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Theorem snidg 3427
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 2056 . 2 𝐴 = 𝐴
2 elsng 3417 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
31, 2mpbiri 161 1 (𝐴𝑉𝐴 ∈ {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  {csn 3402
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 3408
This theorem is referenced by:  snidb  3428  elsn2g  3431  snnzg  3512  snmg  3513  fvunsng  5384  fsnunfv  5390  1stconst  5869  2ndconst  5870  tfr0  5967  tfrlemibxssdm  5971  tfrlemi14d  5977  en1uniel  6314  onunsnss  6385  snon0  6386  supsnti  6408  fseq1p1m1  9057  elfzomin  9163  bj-sels  10393
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