Theorem snsn0non 5763

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 6961). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5764. (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 4779	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4155	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 3881	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4155	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 3881	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2617	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 312	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4140	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 312	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 895	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 5762	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 187	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1475   ∈ wcel 1977  ∅c0 3874  {csn 4125  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by:  onnev  5765  onpsstopbas  31599