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Theorem snsn0non 4640
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 4789). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4897. (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 4327 . . . . 5  |-  { (/) }  e.  _V
21snid 3784 . . . 4  |-  { (/) }  e.  { { (/) } }
3 n0i 3576 . . . 4  |-  ( {
(/) }  e.  { { (/)
} }  ->  -.  { { (/) } }  =  (/) )
42, 3ax-mp 8 . . 3  |-  -.  { { (/) } }  =  (/)
5 0ex 4280 . . . . . . 7  |-  (/)  e.  _V
65snid 3784 . . . . . 6  |-  (/)  e.  { (/)
}
7 n0i 3576 . . . . . 6  |-  ( (/)  e.  { (/) }  ->  -.  {
(/) }  =  (/) )
86, 7ax-mp 8 . . . . 5  |-  -.  { (/)
}  =  (/)
9 eqcom 2389 . . . . 5  |-  ( (/)  =  { (/) }  <->  { (/) }  =  (/) )
108, 9mtbir 291 . . . 4  |-  -.  (/)  =  { (/)
}
115elsnc 3780 . . . 4  |-  ( (/)  e.  { { (/) } }  <->  (/)  =  { (/) } )
1210, 11mtbir 291 . . 3  |-  -.  (/)  e.  { { (/) } }
134, 12pm3.2ni 828 . 2  |-  -.  ( { { (/) } }  =  (/) 
\/  (/)  e.  { { (/)
} } )
14 on0eqel 4639 . 2  |-  ( { { (/) } }  e.  On  ->  ( { { (/)
} }  =  (/)  \/  (/)  e.  { { (/) } } ) )
1513, 14mto 169 1  |-  -.  { { (/) } }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    = wceq 1649    e. wcel 1717   (/)c0 3571   {csn 3757   Oncon0 4522
This theorem is referenced by:  onnev  4898  onpsstopbas  25894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526
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