Theorem snsn0non 6510

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7908). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6511. (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 5402	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4623	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4292	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5265	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4623	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4292	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2770	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 326	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4599	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 326	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 894	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6509	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 200	1 ⊢ ¬ {{∅}} ∈ On

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∅c0 4281  {csn 4588  Oncon0 6383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-tr 5217  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-ord 6386  df-on 6387

This theorem is used by:  onnev  6512  onpsstopbas  37409