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Theorem snsn0non 4511
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 4660). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4769. (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 4197 . . . . 5  |-  { (/) }  e.  _V
21snid 3667 . . . 4  |-  { (/) }  e.  { { (/) } }
3 n0i 3460 . . . 4  |-  ( {
(/) }  e.  { { (/)
} }  ->  -.  { { (/) } }  =  (/) )
42, 3ax-mp 8 . . 3  |-  -.  { { (/) } }  =  (/)
5 0ex 4150 . . . . . . 7  |-  (/)  e.  _V
65snid 3667 . . . . . 6  |-  (/)  e.  { (/)
}
7 n0i 3460 . . . . . 6  |-  ( (/)  e.  { (/) }  ->  -.  {
(/) }  =  (/) )
86, 7ax-mp 8 . . . . 5  |-  -.  { (/)
}  =  (/)
9 eqcom 2285 . . . . 5  |-  ( (/)  =  { (/) }  <->  { (/) }  =  (/) )
108, 9mtbir 290 . . . 4  |-  -.  (/)  =  { (/)
}
115elsnc 3663 . . . 4  |-  ( (/)  e.  { { (/) } }  <->  (/)  =  { (/) } )
1210, 11mtbir 290 . . 3  |-  -.  (/)  e.  { { (/) } }
134, 12pm3.2ni 827 . 2  |-  -.  ( { { (/) } }  =  (/) 
\/  (/)  e.  { { (/)
} } )
14 on0eqel 4510 . 2  |-  ( { { (/) } }  e.  On  ->  ( { { (/)
} }  =  (/)  \/  (/)  e.  { { (/) } } ) )
1513, 14mto 167 1  |-  -.  { { (/) } }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1623    e. wcel 1684   (/)c0 3455   {csn 3640   Oncon0 4392
This theorem is referenced by:  onnev  4770  onpsstopbas  24869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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