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Theorem snsn0non 5001
 Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 6704). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5088. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
snsn0non

Proof of Theorem snsn0non
StepHypRef Expression
1 p0ex 4639 . . . . 5
21snid 4057 . . . 4
3 n0i 3789 . . . 4
42, 3ax-mp 5 . . 3
5 0ex 4582 . . . . . . 7
65snid 4057 . . . . . 6
7 n0i 3789 . . . . . 6
86, 7ax-mp 5 . . . . 5
9 eqcom 2466 . . . . 5
108, 9mtbir 299 . . . 4
115elsnc 4053 . . . 4
1210, 11mtbir 299 . . 3
134, 12pm3.2ni 854 . 2
14 on0eqel 5000 . 2
1513, 14mto 176 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  \/wo 368  =wceq 1395  e.wcel 1818   c0 3784  {csn 4029   con0 4883 This theorem is referenced by:  onnev  5089  onpsstopbas  29895 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 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 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  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 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887
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