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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 7891). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6512. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
snsn0non | ⊢ ¬ {{∅}} ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5442 | . . . . 5 ⊢ {∅} ∈ V | |
2 | 1 | snid 4667 | . . . 4 ⊢ {∅} ∈ {{∅}} |
3 | 2 | n0ii 4349 | . . 3 ⊢ ¬ {{∅}} = ∅ |
4 | 0ex 5313 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 4 | snid 4667 | . . . . . 6 ⊢ ∅ ∈ {∅} |
6 | 5 | n0ii 4349 | . . . . 5 ⊢ ¬ {∅} = ∅ |
7 | eqcom 2742 | . . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅) | |
8 | 6, 7 | mtbir 323 | . . . 4 ⊢ ¬ ∅ = {∅} |
9 | 4 | elsn 4646 | . . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅}) |
10 | 8, 9 | mtbir 323 | . . 3 ⊢ ¬ ∅ ∈ {{∅}} |
11 | 3, 10 | pm3.2ni 880 | . 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}) |
12 | on0eqel 6510 | . 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})) | |
13 | 11, 12 | mto 197 | 1 ⊢ ¬ {{∅}} ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {csn 4631 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: onnev 6513 onpsstopbas 36413 |
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