Theorem notnotsnex 4043

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 4040	. . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2	1	con3i 600	. . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3		snexprc 4042	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
4	2, 3	syl 14	. . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V)
5	4	con3i 600	. 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6		ax-in1 582	. 2 ⊢ ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V)
7	5, 6	ax-mp 7	1 ⊢ ¬ ¬ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1445  Vcvv 2633  {csn 3466

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472

This theorem is referenced by: (None)