![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7331). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6083. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
snsn0non | ⊢ ¬ {{∅}} ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p0ex 5084 | . . . . 5 ⊢ {∅} ∈ V | |
2 | 1 | snid 4430 | . . . 4 ⊢ {∅} ∈ {{∅}} |
3 | 2 | n0ii 4153 | . . 3 ⊢ ¬ {{∅}} = ∅ |
4 | 0ex 5015 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 4 | snid 4430 | . . . . . 6 ⊢ ∅ ∈ {∅} |
6 | 5 | n0ii 4153 | . . . . 5 ⊢ ¬ {∅} = ∅ |
7 | eqcom 2833 | . . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅) | |
8 | 6, 7 | mtbir 315 | . . . 4 ⊢ ¬ ∅ = {∅} |
9 | 4 | elsn 4413 | . . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅}) |
10 | 8, 9 | mtbir 315 | . . 3 ⊢ ¬ ∅ ∈ {{∅}} |
11 | 3, 10 | pm3.2ni 911 | . 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}) |
12 | on0eqel 6081 | . 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})) | |
13 | 11, 12 | mto 189 | 1 ⊢ ¬ {{∅}} ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ∅c0 4145 {csn 4398 Oncon0 5964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-tr 4977 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-ord 5967 df-on 5968 |
This theorem is referenced by: onnev 6084 onpsstopbas 32963 |
Copyright terms: Public domain | W3C validator |