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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 7878). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6498. (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 4667 | . . . 4 ⊢ {∅} ∈ {{∅}} |
3 | 2 | n0ii 4338 | . . 3 ⊢ ¬ {{∅}} = ∅ |
4 | 0ex 5309 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 4 | snid 4667 | . . . . . 6 ⊢ ∅ ∈ {∅} |
6 | 5 | n0ii 4338 | . . . . 5 ⊢ ¬ {∅} = ∅ |
7 | eqcom 2734 | . . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅) | |
8 | 6, 7 | mtbir 322 | . . . 4 ⊢ ¬ ∅ = {∅} |
9 | 4 | elsn 4645 | . . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅}) |
10 | 8, 9 | mtbir 322 | . . 3 ⊢ ¬ ∅ ∈ {{∅}} |
11 | 3, 10 | pm3.2ni 878 | . 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}) |
12 | on0eqel 6496 | . 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})) | |
13 | 11, 12 | mto 196 | 1 ⊢ ¬ {{∅}} ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∅c0 4324 {csn 4630 Oncon0 6372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-tr 5268 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-ord 6375 df-on 6376 |
This theorem is referenced by: onnev 6499 onnevOLD 6500 onpsstopbas 35919 |
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