Theorem notnotsnex 4166

Description: A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)

Assertion

Ref	Expression
notnotsnex	⊢ ¬ ¬ {𝐴} ∈ V

Proof of Theorem notnotsnex

Step	Hyp	Ref	Expression
1		snexg 4163	. . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2	1	con3i 622	. . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3		snexprc 4165	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
4	2, 3	syl 14	. . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V)
5	4	con3i 622	. 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6		pm2.01 606	. 2 ⊢ ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V)
7	5, 6	ax-mp 5	1 ⊢ ¬ ¬ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2136  Vcvv 2726  {csn 3576

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582

This theorem is referenced by: (None)