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Theorem snexg 3926
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 3923 . 2 (A 𝑉 → 𝒫 A V)
2 snsspw 3525 . . 3 {A} ⊆ 𝒫 A
3 ssexg 3886 . . 3 (({A} ⊆ 𝒫 A 𝒫 A V) → {A} V)
42, 3mpan 400 . 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 1390  Vcvv 2551  wss 2911  𝒫 cpw 3350  {csn 3366
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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917
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-in 2918  df-ss 2925  df-pw 3352  df-sn 3372
This theorem is referenced by:  snex  3927  opexg  3954  tpexg  4144  opabex3d  5687  opabex3  5688  mpt2exxg  5772  cnvf1o  5785  brtpos2  5804  tfr0  5875  tfrlemisucaccv  5877  tfrlemibxssdm  5879  tfrlemibfn  5880  xpsnen2g  6232
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