Theorem snsn0non 3788

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 3954). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4068. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
snsn0non	|- -. {{(/)}} e. On

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		p0ex 3495	. . . . 5 |- {(/)} e. _V
2	1	snid 3069	. . . 4 |- {(/)} e. {{(/)}}
3		n0i 2880	. . . 4 |- ({(/)} e. {{(/)}} -> -. {{(/)}} = (/))
4	2, 3	ax-mp 7	. . 3 |- -. {{(/)}} = (/)
5		0ex 3446	. . . . . . 7 |- (/) e. _V
6	5	snid 3069	. . . . . 6 |- (/) e. {(/)}
7		n0i 2880	. . . . . 6 |- ((/) e. {(/)} -> -. {(/)} = (/))
8	6, 7	ax-mp 7	. . . . 5 |- -. {(/)} = (/)
9		eqcom 1886	. . . . 5 |- ((/) = {(/)} <-> {(/)} = (/))
10	8, 9	mtbir 209	. . . 4 |- -. (/) = {(/)}
11	5	elsnc 3065	. . . 4 |- ((/) e. {{(/)}} <-> (/) = {(/)})
12	10, 11	mtbir 209	. . 3 |- -. (/) e. {{(/)}}
13	4, 12	pm3.2ni 640	. 2 |- -. ({{(/)}} = (/) \/ (/) e. {{(/)}})
14		on0eqel 3787	. 2 |- ({{(/)}} e. On -> ({{(/)}} = (/) \/ (/) e. {{(/)}}))
15	13, 14	mto 121	1 |- -. {{(/)}} e. On

Syntax hints:  -. wn 2   \/ wo 239   = wceq 1298   e. wcel 1300  (/)c0 2875  {csn 3044  Oncon0 3657

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

Copyright terms: Public domain