Theorem snexprc 4269

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 3731	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 120	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3		0ex 4210	. 2 ⊢ ∅ ∈ V
4	2, 3	eqeltrdi 2320	1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ∅c0 3491  {csn 3666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4209

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-nul 3492  df-sn 3672

This theorem is referenced by:  notnotsnex  4270