Theorem snexgOLD 3935

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 3936 and new proofs should use snexg 3936 instead. (Contributed by Jim Kingdon, 26-Jan-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of snexg 3936 and then remove it.

Assertion

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

Proof of Theorem snexgOLD

Step	Hyp	Ref	Expression
1		pwexg 3933	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		snsspw 3535	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
3		ssexg 3896	. . 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 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:  snelpwi  3948  snelpw  3949  rext  3951  sspwb  3952  intid  3960  euabex  3961  mss  3962  exss  3963  opexgOLD  3965  opi1  3969  opm  3971  opeqsn  3989  opeqpr  3990  uniop  3992  snnex  4181  op1stb  4209  op1stbg  4210  sucexb  4223  dtruex  4283  relop  4486  elxp4  4808  elxp5  4809  funopg  4934  1stvalg  5769  2ndvalg  5770  fo1st  5784  fo2nd  5785