Theorem snsn0non 6489

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7882). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6490. (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 5396	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4619	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4288	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5260	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4619	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4288	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2767	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 326	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4595	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 326	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 894	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6488	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 200	1 ⊢ ¬ {{∅}} ∈ On

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∅c0 4278  {csn 4584  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366

This theorem is used by:  onnev  6491  onpsstopbas  37311