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Theorem snsn0non 6508
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7892). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6509. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
snsn0non ¬ {{∅}} ∈ On

Proof of Theorem snsn0non
StepHypRef Expression
1 snex 5435 . . . . 5 {∅} ∈ V
21snid 4661 . . . 4 {∅} ∈ {{∅}}
32n0ii 4342 . . 3 ¬ {{∅}} = ∅
4 0ex 5306 . . . . . . 7 ∅ ∈ V
54snid 4661 . . . . . 6 ∅ ∈ {∅}
65n0ii 4342 . . . . 5 ¬ {∅} = ∅
7 eqcom 2743 . . . . 5 (∅ = {∅} ↔ {∅} = ∅)
86, 7mtbir 323 . . . 4 ¬ ∅ = {∅}
94elsn 4640 . . . 4 (∅ ∈ {{∅}} ↔ ∅ = {∅})
108, 9mtbir 323 . . 3 ¬ ∅ ∈ {{∅}}
113, 10pm3.2ni 880 . 2 ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12 on0eqel 6507 . 2 ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
1311, 12mto 197 1 ¬ {{∅}} ∈ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 847   = wceq 1539  wcel 2107  c0 4332  {csn 4625  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  onnev  6510  onpsstopbas  36432
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