Theorem snsn0non 5001

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 6704). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5088. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
snsn0non

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		p0ex 4639	. . . . 5
2	1	snid 4057	. . . 4
3		n0i 3789	. . . 4
4	2, 3	ax-mp 5	. . 3
5		0ex 4582	. . . . . . 7
6	5	snid 4057	. . . . . 6
7		n0i 3789	. . . . . 6
8	6, 7	ax-mp 5	. . . . 5
9		eqcom 2466	. . . . 5
10	8, 9	mtbir 299	. . . 4
11	5	elsnc 4053	. . . 4
12	10, 11	mtbir 299	. . 3
13	4, 12	pm3.2ni 854	. 2
14		on0eqel 5000	. 2
15	13, 14	mto 176	1

Syntax hints:  -.wn 3  \/wo 368  =wceq 1395  e.wcel 1818  c0 3784  {csn 4029  con0 4883

This theorem is referenced by:  onnev  5089  onpsstopbas  29895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887