Theorem snsn0non 6497

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7878). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6498. (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 5435	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4667	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4338	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5309	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4667	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4338	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2734	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 322	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4645	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 322	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 878	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6496	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 196	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ∅c0 4324  {csn 4630  Oncon0 6372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-tr 5268  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-ord 6375  df-on 6376

This theorem is referenced by:  onnev  6499  onnevOLD  6500  onpsstopbas  35919