Theorem snsn0non 6468

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7846). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6469. (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 5395	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4620	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4295	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5256	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4620	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4295	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2768	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 325	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4596	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 325	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 891	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6467	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 199	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 858   = wceq 1559   ∈ wcel 2141  ∅c0 4285  {csn 4581  Oncon0 6342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6345  df-on 6346

This theorem is referenced by:  onnev  6470  onpsstopbas  36754