Theorem notnotsnex 4221

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 4218	. . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2	1	con3i 633	. . . 4 ⊢ (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3		snexprc 4220	. . . 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 2167  Vcvv 2763  {csn 3623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629

This theorem is referenced by: (None)