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Theorem snexg 3908
Description: A singleton whose element exists is a set. The A V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
Assertion
Ref Expression
snexg (A 𝑉 → {A} V)

Proof of Theorem snexg
StepHypRef Expression
1 pwexg 3905 . 2 (A 𝑉 → 𝒫 A V)
2 snsspw 3507 . . 3 {A} ⊆ 𝒫 A
3 ssexg 3868 . . 3 (({A} ⊆ 𝒫 A 𝒫 A V) → {A} V)
42, 3mpan 402 . 2 (𝒫 A V → {A} V)
51, 4syl 14 1 (A 𝑉 → {A} V)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  Vcvv 2533  wss 2892  𝒫 cpw 3332  {csn 3348
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354
This theorem is referenced by:  snex  3909  opexg  3936  tpexg  4127  opabex3d  5669  opabex3  5670  mpt2exxg  5754  cnvf1o  5767  brtpos2  5786  tfrlemisucaccv  5858  tfrlemibxssdm  5860  tfrlemibfn  5861
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