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Mirrors > Home > MPE Home > Th. List > uhgr0 | Structured version Visualization version GIF version |
Description: The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
uhgr0 | ⊢ ∅ ∈ UHGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 6336 | . . 3 ⊢ ∅:∅⟶∅ | |
2 | dm0 5584 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | pw0 4574 | . . . . . 6 ⊢ 𝒫 ∅ = {∅} | |
4 | 3 | difeq1i 3946 | . . . . 5 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅}) |
5 | difid 4178 | . . . . 5 ⊢ ({∅} ∖ {∅}) = ∅ | |
6 | 4, 5 | eqtri 2801 | . . . 4 ⊢ (𝒫 ∅ ∖ {∅}) = ∅ |
7 | 2, 6 | feq23i 6285 | . . 3 ⊢ (∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) ↔ ∅:∅⟶∅) |
8 | 1, 7 | mpbir 223 | . 2 ⊢ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) |
9 | 0ex 5026 | . . 3 ⊢ ∅ ∈ V | |
10 | vtxval0 26387 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
11 | 10 | eqcomi 2786 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
12 | iedgval0 26388 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
13 | 12 | eqcomi 2786 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
14 | 11, 13 | isuhgr 26408 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅}))) |
15 | 9, 14 | ax-mp 5 | . 2 ⊢ (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})) |
16 | 8, 15 | mpbir 223 | 1 ⊢ ∅ ∈ UHGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2106 Vcvv 3397 ∖ cdif 3788 ∅c0 4140 𝒫 cpw 4378 {csn 4397 dom cdm 5355 ⟶wf 6131 ‘cfv 6135 Vtxcvtx 26344 iEdgciedg 26345 UHGraphcuhgr 26404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-slot 16259 df-base 16261 df-edgf 26338 df-vtx 26346 df-iedg 26347 df-uhgr 26406 |
This theorem is referenced by: (None) |
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