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Theorem snsn0non 6499
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7880). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6500. (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 5437 . . . . 5 {∅} ∈ V
21snid 4669 . . . 4 {∅} ∈ {{∅}}
32n0ii 4340 . . 3 ¬ {{∅}} = ∅
4 0ex 5311 . . . . . . 7 ∅ ∈ V
54snid 4669 . . . . . 6 ∅ ∈ {∅}
65n0ii 4340 . . . . 5 ¬ {∅} = ∅
7 eqcom 2735 . . . . 5 (∅ = {∅} ↔ {∅} = ∅)
86, 7mtbir 322 . . . 4 ¬ ∅ = {∅}
94elsn 4647 . . . 4 (∅ ∈ {{∅}} ↔ ∅ = {∅})
108, 9mtbir 322 . . 3 ¬ ∅ ∈ {{∅}}
113, 10pm3.2ni 878 . 2 ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12 on0eqel 6498 . 2 ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
1311, 12mto 196 1 ¬ {{∅}} ∈ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1533  wcel 2098  c0 4326  {csn 4632  Oncon0 6374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378
This theorem is referenced by:  onnev  6501  onnevOLD  6502  onpsstopbas  35947
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