Theorem snexprc 4118

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 3596	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 119	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3		0ex 4063	. 2 ⊢ ∅ ∈ V
4	2, 3	eqeltrdi 2231	1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332   ∈ wcel 1481  Vcvv 2689  ∅c0 3368  {csn 3532

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-nul 3369  df-sn 3538

This theorem is referenced by:  notnotsnex  4119