Theorem snexprc 4229

Description: A singleton whose element is a proper class 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
snexprc	⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)

Proof of Theorem snexprc

Step	Hyp	Ref	Expression
1		snprc 3697	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 120	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3		0ex 4170	. 2 ⊢ ∅ ∈ V
4	2, 3	eqeltrdi 2295	1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1372   ∈ wcel 2175  Vcvv 2771  ∅c0 3459  {csn 3632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-nul 4169

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-nul 3460  df-sn 3638

This theorem is referenced by:  notnotsnex  4230