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Theorem snsn0non 4692
 Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 4841). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4949. (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 4378 . . . . 5
21snid 3833 . . . 4
3 n0i 3625 . . . 4
42, 3ax-mp 8 . . 3
5 0ex 4331 . . . . . . 7
65snid 3833 . . . . . 6
7 n0i 3625 . . . . . 6
86, 7ax-mp 8 . . . . 5
9 eqcom 2437 . . . . 5
108, 9mtbir 291 . . . 4
115elsnc 3829 . . . 4
1210, 11mtbir 291 . . 3
134, 12pm3.2ni 828 . 2
14 on0eqel 4691 . 2
1513, 14mto 169 1
 Colors of variables: wff set class Syntax hints:   wn 3   wo 358   wceq 1652   wcel 1725  c0 3620  csn 3806  con0 4573 This theorem is referenced by:  onnev  4950  onpsstopbas  26172 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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