Theorem snsn0non 6309

Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7584). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6310. (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 5332	. . . . 5 ⊢ {∅} ∈ V
2	1	snid 4601	. . . 4 ⊢ {∅} ∈ {{∅}}
3	2	n0ii 4302	. . 3 ⊢ ¬ {{∅}} = ∅
4		0ex 5211	. . . . . . 7 ⊢ ∅ ∈ V
5	4	snid 4601	. . . . . 6 ⊢ ∅ ∈ {∅}
6	5	n0ii 4302	. . . . 5 ⊢ ¬ {∅} = ∅
7		eqcom 2828	. . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅)
8	6, 7	mtbir 325	. . . 4 ⊢ ¬ ∅ = {∅}
9	4	elsn 4582	. . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅})
10	8, 9	mtbir 325	. . 3 ⊢ ¬ ∅ ∈ {{∅}}
11	3, 10	pm3.2ni 877	. 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})
12		on0eqel 6308	. 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}))
13	11, 12	mto 199	1 ⊢ ¬ {{∅}} ∈ On

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∅c0 4291  {csn 4567  Oncon0 6191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-tr 5173  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-ord 6194  df-on 6195

This theorem is referenced by:  onnev  6311  onpsstopbas  33778