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Theorem snsn0non 6511
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7891). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6512. (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 5442 . . . . 5 {∅} ∈ V
21snid 4667 . . . 4 {∅} ∈ {{∅}}
32n0ii 4349 . . 3 ¬ {{∅}} = ∅
4 0ex 5313 . . . . . . 7 ∅ ∈ V
54snid 4667 . . . . . 6 ∅ ∈ {∅}
65n0ii 4349 . . . . 5 ¬ {∅} = ∅
7 eqcom 2742 . . . . 5 (∅ = {∅} ↔ {∅} = ∅)
86, 7mtbir 323 . . . 4 ¬ ∅ = {∅}
94elsn 4646 . . . 4 (∅ ∈ {{∅}} ↔ ∅ = {∅})
108, 9mtbir 323 . . 3 ¬ ∅ ∈ {{∅}}
113, 10pm3.2ni 880 . 2 ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12 on0eqel 6510 . 2 ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
1311, 12mto 197 1 ¬ {{∅}} ∈ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 847   = wceq 1537  wcel 2106  c0 4339  {csn 4631  Oncon0 6386
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onnev  6513  onpsstopbas  36413
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