Theorem snsn0non 6520

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7907). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6521. (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 5451	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4684	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4366	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5325	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4684	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4366	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2747	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 323	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4663	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 323	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 879	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6519	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 197	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∅c0 4352  {csn 4648  Oncon0 6395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399

This theorem is referenced by:  onnev  6522  onpsstopbas  36396