Theorem snsn0non 6082

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7331). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6083. (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 5084	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4430	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4153	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5015	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4430	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4153	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2833	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 315	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4413	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 315	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 911	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6081	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 189	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 880   = wceq 1658   ∈ wcel 2166  ∅c0 4145  {csn 4398  Oncon0 5964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-tr 4977  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-ord 5967  df-on 5968

This theorem is referenced by:  onnev  6084  onpsstopbas  32963