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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

Proof of Theorem snsn0non
StepHypRef Expression
1 p0ex 4643 . . . . 5
21snid 4060 . . . 4
3 n0i 3798 . . . 4
42, 3ax-mp 5 . . 3
5 0ex 4587 . . . . . . 7
65snid 4060 . . . . . 6
7 n0i 3798 . . . . . 6
86, 7ax-mp 5 . . . . 5
9 eqcom 2466 . . . . 5
108, 9mtbir 299 . . . 4
115elsnc 4056 . . . 4
1210, 11mtbir 299 . . 3
134, 12pm3.2ni 854 . 2
14 on0eqel 5004 . 2
1513, 14mto 176 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wo 368   wceq 1395   wcel 1819  c0 3793  csn 4032  con0 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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