Theorem snsn0non 6489

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7867). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6490. (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 5412	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4629	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4297	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5271	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4629	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4297	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2770	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 326	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4605	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 326	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 893	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6488	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 200	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∅c0 4287  {csn 4590  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366

This theorem is referenced by:  onnev  6491  onpsstopbas  36922