Theorem snexprc 4282

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 3738	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 120	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3		0ex 4221	. 2 ⊢ ∅ ∈ V
4	2, 3	eqeltrdi 2322	1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ∈ wcel 2202  Vcvv 2803  ∅c0 3496  {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-nul 3497  df-sn 3679

This theorem is referenced by:  notnotsnex  4283