Theorem snsn0non 6432

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7800). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6433. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
snsn0non	⊢ ¬ {{∅}} ∈ On

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		snex 5374	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4615	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4293	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5245	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4615	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4293	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2738	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 323	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4591	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 323	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 880	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6431	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 197	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∅c0 4283  {csn 4576  Oncon0 6306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310

This theorem is referenced by:  onnev  6434  onpsstopbas  36463