Theorem snsn0non 6488

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7870). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6489. (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 5408	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4626	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4292	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5268	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4626	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4292	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2769	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 326	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4602	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 326	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 894	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6487	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 200	1 ⊢ ¬ {{∅}} ∈ On

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∅c0 4282  {csn 4587  Oncon0 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365

This theorem is used by:  onnev  6490  onpsstopbas  37057