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Theorem snsn0non 5810
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7023). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5811. (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 p0ex 4818 . . . . 5 {∅} ∈ V
21snid 4184 . . . 4 {∅} ∈ {{∅}}
32n0ii 3903 . . 3 ¬ {{∅}} = ∅
4 0ex 4755 . . . . . . 7 ∅ ∈ V
54snid 4184 . . . . . 6 ∅ ∈ {∅}
65n0ii 3903 . . . . 5 ¬ {∅} = ∅
7 eqcom 2628 . . . . 5 (∅ = {∅} ↔ {∅} = ∅)
86, 7mtbir 313 . . . 4 ¬ ∅ = {∅}
94elsn 4168 . . . 4 (∅ ∈ {{∅}} ↔ ∅ = {∅})
108, 9mtbir 313 . . 3 ¬ ∅ ∈ {{∅}}
113, 10pm3.2ni 898 . 2 ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12 on0eqel 5809 . 2 ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
1311, 12mto 188 1 ¬ {{∅}} ∈ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383   = wceq 1480  wcel 1987  c0 3896  {csn 4153  Oncon0 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691
This theorem is referenced by:  onnev  5812  onpsstopbas  32106
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