Theorem snsn0non 6008

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7236). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6009. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
snsn0non	⊢ ¬ {{∅}} ∈ On

Proof of Theorem snsn0non

Step	Hyp	Ref	Expression
1		p0ex 5003	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4354	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4066	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 4943	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4354	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4066	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2768	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 312	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4337	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 312	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 935	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6007	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 188	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1632   ∈ wcel 2140  ∅c0 4059  {csn 4322  Oncon0 5885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-tr 4906  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-ord 5888  df-on 5889

This theorem is referenced by:  onnev  6010  onpsstopbas  32757