Theorem notnotsnex 4277

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 4274	. . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2	1	con3i 637	. . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3		snexprc 4276	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
4	2, 3	syl 14	. . 3 ⊢ (¬ {𝐴} ∈ V → {𝐴} ∈ V)
5	4	con3i 637	. 2 ⊢ (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6		pm2.01 621	. 2 ⊢ ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V)
7	5, 6	ax-mp 5	1 ⊢ ¬ ¬ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2202  Vcvv 2802  {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675

This theorem is referenced by: (None)