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Theorem snsn0non 3115

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 3126). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 3231.

Assertion

Ref	Expression
snsn0non

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Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		0ex 2701	. . . . 5
2	1	snnz 2449	. . . 4
3	1	elsnc 2421	. . . . 5
4		eqcom 1469	. . . . 5
5	3, 4	bitr 173	. . . 4
6	2, 5	nemtbir 1633	. . 3
7	1	snid 2425	. . . 4
8		ssel 2053	. . . 4
9	7, 8	mpi 44	. . 3
10	6, 9	mto 106	. 2
11		p0ex 2760	. . . 4
12	11	snid 2425	. . 3
13		onelsst 2990	. . 3
14	12, 13	mpi 44	. 2
15	10, 14	mto 106	1

Colors of variables: wff set class

Syntax hints:   wn 2   wceq 953   wcel 955   wss 2037  c0 2270  csn 2399  con0 2938

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-tr 2671  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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