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Theorem snsn0non 6282
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7559). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6283. (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 5305 . . . . 5 {∅} ∈ V
21snid 4574 . . . 4 {∅} ∈ {{∅}}
32n0ii 4275 . . 3 ¬ {{∅}} = ∅
4 0ex 5184 . . . . . . 7 ∅ ∈ V
54snid 4574 . . . . . 6 ∅ ∈ {∅}
65n0ii 4275 . . . . 5 ¬ {∅} = ∅
7 eqcom 2828 . . . . 5 (∅ = {∅} ↔ {∅} = ∅)
86, 7mtbir 326 . . . 4 ¬ ∅ = {∅}
94elsn 4555 . . . 4 (∅ ∈ {{∅}} ↔ ∅ = {∅})
108, 9mtbir 326 . . 3 ¬ ∅ ∈ {{∅}}
113, 10pm3.2ni 878 . 2 ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12 on0eqel 6281 . 2 ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
1311, 12mto 200 1 ¬ {{∅}} ∈ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 844   = wceq 1538  wcel 2115  c0 4266  {csn 4540  Oncon0 6164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-tr 5146  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6167  df-on 6168
This theorem is referenced by:  onnev  6284  onpsstopbas  33785
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