Theorem snsn0non 6475

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7839). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6476. (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 5421	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4655	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4329	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5297	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4655	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4329	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2738	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 322	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4634	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 322	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 879	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6474	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 196	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∅c0 4315  {csn 4619  Oncon0 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6353  df-on 6354

This theorem is referenced by:  onnev  6477  onnevOLD  6478  onpsstopbas  35103