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Theorem snsn0non 5005
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 6703). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5092. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
snsn0non  |-  -.  { { (/) } }  e.  On

Proof of Theorem snsn0non
StepHypRef Expression
1 p0ex 4643 . . . . 5  |-  { (/) }  e.  _V
21snid 4060 . . . 4  |-  { (/) }  e.  { { (/) } }
3 n0i 3798 . . . 4  |-  ( {
(/) }  e.  { { (/)
} }  ->  -.  { { (/) } }  =  (/) )
42, 3ax-mp 5 . . 3  |-  -.  { { (/) } }  =  (/)
5 0ex 4587 . . . . . . 7  |-  (/)  e.  _V
65snid 4060 . . . . . 6  |-  (/)  e.  { (/)
}
7 n0i 3798 . . . . . 6  |-  ( (/)  e.  { (/) }  ->  -.  {
(/) }  =  (/) )
86, 7ax-mp 5 . . . . 5  |-  -.  { (/)
}  =  (/)
9 eqcom 2466 . . . . 5  |-  ( (/)  =  { (/) }  <->  { (/) }  =  (/) )
108, 9mtbir 299 . . . 4  |-  -.  (/)  =  { (/)
}
115elsnc 4056 . . . 4  |-  ( (/)  e.  { { (/) } }  <->  (/)  =  { (/) } )
1210, 11mtbir 299 . . 3  |-  -.  (/)  e.  { { (/) } }
134, 12pm3.2ni 854 . 2  |-  -.  ( { { (/) } }  =  (/) 
\/  (/)  e.  { { (/)
} } )
14 on0eqel 5004 . 2  |-  ( { { (/) } }  e.  On  ->  ( { { (/)
} }  =  (/)  \/  (/)  e.  { { (/) } } ) )
1513, 14mto 176 1  |-  -.  { { (/) } }  e.  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1395    e. wcel 1819   (/)c0 3793   {csn 4032   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  onnev  5093  onpsstopbas  30100
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