Theorem snexprc 4172

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 3648	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 119	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3		0ex 4116	. 2 ⊢ ∅ ∈ V
4	2, 3	eqeltrdi 2261	1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1348   ∈ wcel 2141  Vcvv 2730  ∅c0 3414  {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3589

This theorem is referenced by:  notnotsnex  4173