Theorem snsn0non 6370

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7691). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6371. (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		snex 5349	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4594	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4267	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5226	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4594	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4267	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2745	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 322	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4573	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 322	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 877	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6369	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 196	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∅c0 4253  {csn 4558  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  onnev  6372  onnevOLD  6373  onpsstopbas  34546