Theorem snsn0non 3125

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 3136). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 3241.

Assertion

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

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		0ex 2711	. . . . 5 |- (/) e. V
2	1	snnz 2458	. . . 4 |- {(/)} =/= (/)
3	1	elsnc 2431	. . . . 5 |- ((/) e. {{(/)}} <-> (/) = {(/)})
4		eqcom 1477	. . . . 5 |- ((/) = {(/)} <-> {(/)} = (/))
5	3, 4	bitr 173	. . . 4 |- ((/) e. {{(/)}} <-> {(/)} = (/))
6	2, 5	nemtbir 1641	. . 3 |- -. (/) e. {{(/)}}
7	1	snid 2435	. . . 4 |- (/) e. {(/)}
8		ssel 2063	. . . 4 |- ({(/)} (_ {{(/)}} -> ((/) e. {(/)} -> (/) e. {{(/)}}))
9	7, 8	mpi 44	. . 3 |- ({(/)} (_ {{(/)}} -> (/) e. {{(/)}})
10	6, 9	mto 106	. 2 |- -. {(/)} (_ {{(/)}}
11		p0ex 2770	. . . 4 |- {(/)} e. V
12	11	snid 2435	. . 3 |- {(/)} e. {{(/)}}
13		onelsst 3000	. . 3 |- ({{(/)}} e. On -> ({(/)} e. {{(/)}} -> {(/)} (_ {{(/)}}))
14	12, 13	mpi 44	. 2 |- ({{(/)}} e. On -> {(/)} (_ {{(/)}})
15	10, 14	mto 106	1 |- -. {{(/)}} e. On

Syntax hints:  -. wn 2   = wceq 956   e. wcel 958   (_ wss 2047  (/)c0 2280  {csn 2409  Oncon0 2948

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-tr 2681  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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