ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snexg GIF version

Theorem snexg 4076
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 4072 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 snsspw 3659 . . 3 {𝐴} ⊆ 𝒫 𝐴
3 ssexg 4035 . . 3 (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V)
42, 3mpan 418 . 2 (𝒫 𝐴 ∈ V → {𝐴} ∈ V)
51, 4syl 14 1 (𝐴𝑉 → {𝐴} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  Vcvv 2658  wss 3039  𝒫 cpw 3478  {csn 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501
This theorem is referenced by:  snex  4077  notnotsnex  4079  exmidsssnc  4094  snelpwi  4102  opexg  4118  opm  4124  tpexg  4333  op1stbg  4368  sucexb  4381  elxp4  4994  elxp5  4995  opabex3d  5985  opabex3  5986  1stvalg  6006  2ndvalg  6007  mpoexxg  6074  cnvf1o  6088  brtpos2  6114  tfr0dm  6185  tfrlemisucaccv  6188  tfrlemibxssdm  6190  tfrlemibfn  6191  tfr1onlemsucaccv  6204  tfr1onlembxssdm  6206  tfr1onlembfn  6207  tfrcllemsucaccv  6217  tfrcllembxssdm  6219  tfrcllembfn  6220  fvdiagfn  6553  ixpsnf1o  6596  mapsnf1o  6597  xpsnen2g  6689  zfz1isolem1  10523  climconst2  11000  ennnfonelemp1  11814  setsvalg  11884  setsex  11886  setsslid  11904  strle1g  11944  1strbas  11953
  Copyright terms: Public domain W3C validator