Theorem snsn0non 6483

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7856). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6484. (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 5424	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4659	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4331	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5300	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4659	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4331	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2733	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 323	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4638	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 323	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 877	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6482	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 196	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1533   ∈ wcel 2098  ∅c0 4317  {csn 4623  Oncon0 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362

This theorem is referenced by:  onnev  6485  onnevOLD  6486  onpsstopbas  35823