Theorem snsn0non 5846

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7069). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5847. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

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

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		p0ex 4853	. . . . 5 |- { (/) } e. _V
2	1	snid 4208	. . . 4 |- { (/) } e. { { (/) } }
3	2	n0ii 3922	. . 3 |- -. { { (/) } } = (/)
4		0ex 4790	. . . . . . 7 |- (/) e. _V
5	4	snid 4208	. . . . . 6 |- (/) e. { (/) }
6	5	n0ii 3922	. . . . 5 |- -. { (/) } = (/)
7		eqcom 2629	. . . . 5 |- ( (/) = { (/) } <-> { (/) } = (/) )
8	6, 7	mtbir 313	. . . 4 |- -. (/) = { (/) }
9	4	elsn 4192	. . . 4 |- ( (/) e. { { (/) } } <-> (/) = { (/) } )
10	8, 9	mtbir 313	. . 3 |- -. (/) e. { { (/) } }
11	3, 10	pm3.2ni 899	. 2 |- -. ( { { (/) } } = (/) \/ (/) e. { { (/) } } )
12		on0eqel 5845	. 2 |- ( { { (/) } } e. On -> ( { { (/) } } = (/) \/ (/) e. { { (/) } } ) )
13	11, 12	mto 188	1 |- -. { { (/) } } e. On

Syntax hints:   -. wn 3   \/ wo 383   = wceq 1483   e. wcel 1990   (/)c0 3915   {csn 4177   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  onnev  5848  onpsstopbas  32429