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Theorem snexg 4025
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 (𝐴𝑉 → {𝐴} ∈ V)

Proof of Theorem snexg
StepHypRef Expression
1 pwexg 4021 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 snsspw 3614 . . 3 {𝐴} ⊆ 𝒫 𝐴
3 ssexg 3984 . . 3 (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
42, 3mpan 416 . 2 (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
51, 4syl 14 1 (𝐴𝑉 → {𝐴} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1439  Vcvv 2620  wss 3000  𝒫 cpw 3433  {csn 3450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456
This theorem is referenced by:  snex  4026  notnotsnex  4028  snelpwi  4048  opexg  4064  opm  4070  tpexg  4279  op1stbg  4314  sucexb  4327  elxp4  4931  elxp5  4932  opabex3d  5906  opabex3  5907  1stvalg  5927  2ndvalg  5928  mpt2exxg  5991  cnvf1o  6004  brtpos2  6030  tfr0dm  6101  tfrlemisucaccv  6104  tfrlemibxssdm  6106  tfrlemibfn  6107  tfr1onlemsucaccv  6120  tfr1onlembxssdm  6122  tfr1onlembfn  6123  tfrcllemsucaccv  6133  tfrcllembxssdm  6135  tfrcllembfn  6136  fvdiagfn  6464  ixpsnf1o  6507  mapsnf1o  6508  xpsnen2g  6599  zfz1isolem1  10306  climconst2  10740  setsvalg  11585  setsex  11587  setsslid  11605  strle1g  11645  1strbas  11654
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