Theorem snsn0non 6308

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7583). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6309. (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 5331	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4600	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4301	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5210	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4600	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4301	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2828	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 325	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4581	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 325	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 877	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6307	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 199	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1533   ∈ wcel 2110  ∅c0 4290  {csn 4566  Oncon0 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-tr 5172  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-on 6194

This theorem is referenced by:  onnev  6310  onpsstopbas  33778