Theorem snsn0non 6462

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7849). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6463. (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 5394	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4629	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4309	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5265	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4629	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4309	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2737	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 323	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4607	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 323	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 880	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6461	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 197	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∅c0 4299  {csn 4592  Oncon0 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339

This theorem is referenced by:  onnev  6464  onpsstopbas  36425