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Theorem snexg 3936
 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 3933 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 snsspw 3535 . . 3 {𝐴} ⊆ 𝒫 𝐴
3 ssexg 3896 . . 3 (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
42, 3mpan 400 . 2 (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
51, 4syl 14 1 (𝐴𝑉 → {𝐴} ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359  {csn 3375 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381 This theorem is referenced by:  snex  3937  opexg  3964  tpexg  4179  opabex3d  5748  opabex3  5749  mpt2exxg  5833  cnvf1o  5846  brtpos2  5866  tfr0  5937  tfrlemisucaccv  5939  tfrlemibxssdm  5941  tfrlemibfn  5942  xpsnen2g  6303  iseqid3  9245  climconst2  9812
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