Theorem snsn0non 2371

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 2377). It can be used to represent an "undefined" value for a partial operation on natural numbers or ordinals. See also onxpdisj 2476.

Assertion

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

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		0ex 1745	. . . . 5 |- (/) e. V
2	1	snnz 1846	. . . 4 |- -. {(/)} = (/)
3	1	elsnc 1826	. . . . 5 |- ((/) e. {{(/)}} <-> (/) = {(/)})
4		cleqcom 1103	. . . . 5 |- ((/) = {(/)} <-> {(/)} = (/))
5	3, 4	bitr 151	. . . 4 |- ((/) e. {{(/)}} <-> {(/)} = (/))
6	2, 5	mtbir 167	. . 3 |- -. (/) e. {{(/)}}
7	1	snid 1830	. . . 4 |- (/) e. {(/)}
8		ssel 1502	. . . 4 |- ({(/)} (_ {{(/)}} -> ((/) e. {(/)} -> (/) e. {{(/)}}))
9	7, 8	mpi 44	. . 3 |- ({(/)} (_ {{(/)}} -> (/) e. {{(/)}})
10	6, 9	mto 93	. 2 |- -. {(/)} (_ {{(/)}}
11		p0ex 1885	. . . 4 |- {(/)} e. V
12	11	snid 1830	. . 3 |- {(/)} e. {{(/)}}
13		onelsst 2255	. . 3 |- ({{(/)}} e. On -> ({(/)} e. {{(/)}} -> {(/)} (_ {{(/)}}))
14	12, 13	mpi 44	. 2 |- ({{(/)}} e. On -> {(/)} (_ {{(/)}})
15	10, 14	mto 93	1 |- -. {{(/)}} e. On

Syntax hints:  -. wn 1   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707  {csn 1808  Oncon0 2199

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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