Theorem snsn0non 6476

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7854). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6477. (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 5401	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4624	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4298	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5262	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4624	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4298	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2772	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 326	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4600	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 326	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 893	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6475	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 200	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 860   = wceq 1563   ∈ wcel 2145  ∅c0 4288  {csn 4585  Oncon0 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-ord 6353  df-on 6354

This theorem is referenced by:  onnev  6478  onpsstopbas  36803