Theorem snsn0non 6443

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7817). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6444. (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 5375	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4601	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4278	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5236	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4601	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4278	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2747	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 324	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4577	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 324	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 886	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6442	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 198	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∅c0 4268  {csn 4562  Oncon0 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6320  df-on 6321

This theorem is referenced by:  onnev  6445  onpsstopbas  36665