Theorem snsn0non 6443

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7814). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6444. (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 5376	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4607	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4284	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5242	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4607	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4284	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2744	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 323	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4583	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 323	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 881	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6442	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 197	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∅c0 4274  {csn 4568  Oncon0 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321

This theorem is referenced by:  onnev  6445  onpsstopbas  36628