Theorem notnotsnex 4216

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 4213	. . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2	1	con3i 633	. . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3		snexprc 4215	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
4	2, 3	syl 14	. . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V)
5	4	con3i 633	. 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6		pm2.01 617	. 2 ⊢ ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V)
7	5, 6	ax-mp 5	1 ⊢ ¬ ¬ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2164  Vcvv 2760  {csn 3618

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624

This theorem is referenced by: (None)