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

Theorem snexg 3888
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 3885 . 2 (A 𝑉 → 𝒫 A V)
2 snsspw 3487 . . 3 {A} ⊆ 𝒫 A
3 ssexg 3848 . . 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 2533  wss 2895  𝒫 cpw 3311  {csn 3327
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333
This theorem is referenced by:  snex  3889  opabex3d  5637  opabex3  5638  mpt2exxg  5722  cnvf1o  5735  brtpos2  5754  tfrlemisucaccv  5825  tfrlemibxssdm  5827  tfrlemibfn  5828
  Copyright terms: Public domain W3C validator