MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsn0non Unicode version

Theorem snsn0non 4527
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 4676). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4785. (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 4213 . . . . 5  |-  { (/) }  e.  _V
21snid 3680 . . . 4  |-  { (/) }  e.  { { (/) } }
3 n0i 3473 . . . 4  |-  ( {
(/) }  e.  { { (/)
} }  ->  -.  { { (/) } }  =  (/) )
42, 3ax-mp 8 . . 3  |-  -.  { { (/) } }  =  (/)
5 0ex 4166 . . . . . . 7  |-  (/)  e.  _V
65snid 3680 . . . . . 6  |-  (/)  e.  { (/)
}
7 n0i 3473 . . . . . 6  |-  ( (/)  e.  { (/) }  ->  -.  {
(/) }  =  (/) )
86, 7ax-mp 8 . . . . 5  |-  -.  { (/)
}  =  (/)
9 eqcom 2298 . . . . 5  |-  ( (/)  =  { (/) }  <->  { (/) }  =  (/) )
108, 9mtbir 290 . . . 4  |-  -.  (/)  =  { (/)
}
115elsnc 3676 . . . 4  |-  ( (/)  e.  { { (/) } }  <->  (/)  =  { (/) } )
1210, 11mtbir 290 . . 3  |-  -.  (/)  e.  { { (/) } }
134, 12pm3.2ni 827 . 2  |-  -.  ( { { (/) } }  =  (/) 
\/  (/)  e.  { { (/)
} } )
14 on0eqel 4526 . 2  |-  ( { { (/) } }  e.  On  ->  ( { { (/)
} }  =  (/)  \/  (/)  e.  { { (/) } } ) )
1513, 14mto 167 1  |-  -.  { { (/) } }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1632    e. wcel 1696   (/)c0 3468   {csn 3653   Oncon0 4408
This theorem is referenced by:  onnev  4786  onpsstopbas  24941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
  Copyright terms: Public domain W3C validator