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Theorem snidg 3392
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 (A 𝑉A {A})

Proof of Theorem snidg
StepHypRef Expression
1 eqid 2037 . 2 A = A
2 elsncg 3389 . 2 (A 𝑉 → (A {A} ↔ A = A))
31, 2mpbiri 157 1 (A 𝑉A {A})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sn 3373
This theorem is referenced by:  snidb  3393  elsnc2g  3396  snnzg  3476  snmg  3477  fvunsng  5300  fsnunfv  5306  1stconst  5784  2ndconst  5785  tfr0  5878  tfrlemibxssdm  5882  tfrlemi14d  5888  en1uniel  6220  fseq1p1m1  8726  elfzomin  8832  bj-sels  9369
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