Theorem snsn0nonOLD 3789

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.

Assertion

Ref	Expression
snsn0nonOLD	|- -. {{(/)}} e. On

Proof of Theorem snsn0nonOLD

Step	Hyp	Ref	Expression
1		0ex 3446	. . . . 5 |- (/) e. _V
2	1	snnz 3119	. . . 4 |- {(/)} =/= (/)
3	1	elsnc 3065	. . . . 5 |- ((/) e. {{(/)}} <-> (/) = {(/)})
4		eqcom 1886	. . . . 5 |- ((/) = {(/)} <-> {(/)} = (/))
5	3, 4	bitri 190	. . . 4 |- ((/) e. {{(/)}} <-> {(/)} = (/))
6	2, 5	nemtbir 2099	. . 3 |- -. (/) e. {{(/)}}
7	1	snid 3069	. . . 4 |- (/) e. {(/)}
8		ssel 2615	. . . 4 |- ({(/)} C_ {{(/)}} -> ((/) e. {(/)} -> (/) e. {{(/)}}))
9	7, 8	mpi 55	. . 3 |- ({(/)} C_ {{(/)}} -> (/) e. {{(/)}})
10	6, 9	mto 121	. 2 |- -. {(/)} C_ {{(/)}}
11		p0ex 3495	. . . 4 |- {(/)} e. _V
12	11	snid 3069	. . 3 |- {(/)} e. {{(/)}}
13		onelss 3705	. . 3 |- ({{(/)}} e. On -> ({(/)} e. {{(/)}} -> {(/)} C_ {{(/)}}))
14	12, 13	mpi 55	. 2 |- ({{(/)}} e. On -> {(/)} C_ {{(/)}})
15	10, 14	mto 121	1 |- -. {{(/)}} e. On

Syntax hints:  -. wn 2   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  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-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-uni 3178  df-tr 3412  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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