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Theorem snexg 3909
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 3906 . 2 (A 𝑉 → 𝒫 A V)
2 snsspw 3508 . . 3 {A} ⊆ 𝒫 A
3 ssexg 3869 . . 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 1375  Vcvv 2534  wss 2893  𝒫 cpw 3333  {csn 3349
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 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355
This theorem is referenced by:  snex  3910  opexg  3937  tpexg  4127  opabex3d  5668  opabex3  5669  mpt2exxg  5753  cnvf1o  5766  brtpos2  5785  tfrlemisucaccv  5855  tfrlemibxssdm  5857  tfrlemibfn  5858
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