Theorem snexgOLD 3926

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. This is a special case of snexg 3927 and new proofs should use snexg 3927 instead. (Contributed by Jim Kingdon, 26-Jan-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of snexg 3927 and then remove it.

Assertion

Ref	Expression
snexgOLD	⊢ (A ∈ V → {A} ∈ V)

Proof of Theorem snexgOLD

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:  snelpwi  3939  snelpw  3940  rext  3942  sspwb  3943  intid  3951  euabex  3952  mss  3953  exss  3954  opexgOLD  3956  opi1  3960  opm  3962  opeqsn  3980  opeqpr  3981  uniop  3983  snnex  4147  op1stb  4175  op1stbg  4176  sucexb  4189  dtruex  4237  relop  4429  elxp4  4751  elxp5  4752  funopg  4877  1stvalg  5711  2ndvalg  5712  fo1st  5726  fo2nd  5727