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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 6961). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5764. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
snsn0non | ⊢ ¬ {{∅}} ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p0ex 4779 | . . . . 5 ⊢ {∅} ∈ V | |
2 | 1 | snid 4155 | . . . 4 ⊢ {∅} ∈ {{∅}} |
3 | 2 | n0ii 3881 | . . 3 ⊢ ¬ {{∅}} = ∅ |
4 | 0ex 4718 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 4 | snid 4155 | . . . . . 6 ⊢ ∅ ∈ {∅} |
6 | 5 | n0ii 3881 | . . . . 5 ⊢ ¬ {∅} = ∅ |
7 | eqcom 2617 | . . . . 5 ⊢ (∅ = {∅} ↔ {∅} = ∅) | |
8 | 6, 7 | mtbir 312 | . . . 4 ⊢ ¬ ∅ = {∅} |
9 | 4 | elsn 4140 | . . . 4 ⊢ (∅ ∈ {{∅}} ↔ ∅ = {∅}) |
10 | 8, 9 | mtbir 312 | . . 3 ⊢ ¬ ∅ ∈ {{∅}} |
11 | 3, 10 | pm3.2ni 895 | . 2 ⊢ ¬ ({{∅}} = ∅ ∨ ∅ ∈ {{∅}}) |
12 | on0eqel 5762 | . 2 ⊢ ({{∅}} ∈ On → ({{∅}} = ∅ ∨ ∅ ∈ {{∅}})) | |
13 | 11, 12 | mto 187 | 1 ⊢ ¬ {{∅}} ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∅c0 3874 {csn 4125 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: onnev 5765 onpsstopbas 31599 |
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