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| Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7892). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6509. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) | 
| Ref | Expression | 
|---|---|
| snsn0non | ⊢ ¬ {{∅}} ∈ On | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | snex 5435 | . . . . 5 ⊢ {∅} ∈ V | |
| 2 | 1 | snid 4661 | . . . 4 ⊢ {∅} ∈ {{∅}} | 
| 3 | 2 | n0ii 4342 | . . 3 ⊢ ¬ {{∅}} = ∅ | 
| 4 | 0ex 5306 | . . . . . . 7 ⊢ ∅ ∈ V | |
| 5 | 4 | snid 4661 | . . . . . 6 ⊢ ∅ ∈ {∅} | 
| 6 | 5 | n0ii 4342 | . . . . 5 ⊢ ¬ {∅} = ∅ | 
| 7 | eqcom 2743 | . . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅) | |
| 8 | 6, 7 | mtbir 323 | . . . 4 ⊢ ¬ ∅ = {∅} | 
| 9 | 4 | elsn 4640 | . . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅}) | 
| 10 | 8, 9 | mtbir 323 | . . 3 ⊢ ¬ ∅ ∈ {{∅}} | 
| 11 | 3, 10 | pm3.2ni 880 | . 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}) | 
| 12 | on0eqel 6507 | . 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})) | |
| 13 | 11, 12 | mto 197 | 1 ⊢ ¬ {{∅}} ∈ On | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∅c0 4332 {csn 4625 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: onnev 6510 onpsstopbas 36432 | 
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