Theorem snexg 3927

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

Step	Hyp	Ref	Expression
1		pwexg 3924	. 2 ⊢ (A ∈ 𝑉 → 𝒫 A ∈ V)
2		snsspw 3526	. . 3 ⊢ {A} ⊆ 𝒫 A
3		ssexg 3887	. . 3 ⊢ (({A} ⊆ 𝒫 A ∧ 𝒫 A ∈ V) → {A} ∈ V)
4	2, 3	mpan 400	. 2 ⊢ (𝒫 A ∈ V → {A} ∈ V)
5	1, 4	syl 14	1 ⊢ (A ∈ 𝑉 → {A} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  𝒫 cpw 3351  {csn 3367

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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

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 3353  df-sn 3373

This theorem is referenced by:  snex  3928  opexg  3955  tpexg  4145  opabex3d  5690  opabex3  5691  mpt2exxg  5775  cnvf1o  5788  brtpos2  5807  tfr0  5878  tfrlemisucaccv  5880  tfrlemibxssdm  5882  tfrlemibfn  5883  xpsnen2g  6239