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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 6322 | . . 3 ⊢ ∅:∅⟶∅ | |
2 | dm0 5570 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | pw0 4560 | . . . . . 6 ⊢ 𝒫 ∅ = {∅} | |
4 | 3 | difeq1i 3950 | . . . . 5 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅}) |
5 | difid 4177 | . . . . 5 ⊢ ({∅} ∖ {∅}) = ∅ | |
6 | 4, 5 | eqtri 2848 | . . . 4 ⊢ (𝒫 ∅ ∖ {∅}) = ∅ |
7 | 2, 6 | feq23i 6271 | . . 3 ⊢ (∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) ↔ ∅:∅⟶∅) |
8 | 1, 7 | mpbir 223 | . 2 ⊢ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) |
9 | 0ex 5013 | . . 3 ⊢ ∅ ∈ V | |
10 | vtxval0 26336 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
11 | 10 | eqcomi 2833 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
12 | iedgval0 26337 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
13 | 12 | eqcomi 2833 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
14 | 11, 13 | isuhgr 26357 | . . 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 2166 Vcvv 3413 ∖ cdif 3794 ∅c0 4143 𝒫 cpw 4377 {csn 4396 dom cdm 5341 ⟶wf 6118 ‘cfv 6122 Vtxcvtx 26293 iEdgciedg 26294 UHGraphcuhgr 26353 |
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-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-fv 6130 df-slot 16225 df-base 16227 df-edgf 26287 df-vtx 26295 df-iedg 26296 df-uhgr 26355 |
This theorem is referenced by: (None) |
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