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Mirrors > Home > MPE Home > Th. List > snsn0non | Structured version Visualization version GIF version |
Description: The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7564). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6278. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
snsn0non | ⊢ ¬ {{∅}} ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5297 | . . . . 5 ⊢ {∅} ∈ V | |
2 | 1 | snid 4561 | . . . 4 ⊢ {∅} ∈ {{∅}} |
3 | 2 | n0ii 4252 | . . 3 ⊢ ¬ {{∅}} = ∅ |
4 | 0ex 5175 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 4 | snid 4561 | . . . . . 6 ⊢ ∅ ∈ {∅} |
6 | 5 | n0ii 4252 | . . . . 5 ⊢ ¬ {∅} = ∅ |
7 | eqcom 2805 | . . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅) | |
8 | 6, 7 | mtbir 326 | . . . 4 ⊢ ¬ ∅ = {∅} |
9 | 4 | elsn 4540 | . . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅}) |
10 | 8, 9 | mtbir 326 | . . 3 ⊢ ¬ ∅ ∈ {{∅}} |
11 | 3, 10 | pm3.2ni 878 | . 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}) |
12 | on0eqel 6276 | . 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})) | |
13 | 11, 12 | mto 200 | 1 ⊢ ¬ {{∅}} ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∅c0 4243 {csn 4525 Oncon0 6159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 |
This theorem is referenced by: onnev 6279 onnevOLD 6280 onpsstopbas 33891 |
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