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Mirrors > Home > MPE Home > Th. List > uhgr0v0e | Structured version Visualization version GIF version |
Description: The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.) |
Ref | Expression |
---|---|
uhgr0v0e.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgr0v0e.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
uhgr0v0e | ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgr0v0e.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | eqeq1i 2782 | . . . . 5 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
3 | uhgr0vb 26420 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) | |
4 | 3 | biimpd 221 | . . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)) |
5 | 4 | ex 403 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))) |
6 | 2, 5 | syl5bi 234 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))) |
7 | 6 | pm2.43a 54 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅)) |
8 | 7 | imp 397 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅) |
9 | uhgr0v0e.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
10 | 9 | eqeq1i 2782 | . . . 4 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅) |
11 | uhgriedg0edg0 26475 | . . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
12 | 10, 11 | syl5bb 275 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
13 | 12 | adantr 474 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
14 | 8, 13 | mpbird 249 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∅c0 4140 ‘cfv 6135 Vtxcvtx 26344 iEdgciedg 26345 Edgcedg 26395 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 ax-un 7226 |
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-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 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-edg 26396 df-uhgr 26406 |
This theorem is referenced by: uhgr0vsize0 26586 uhgr0vusgr 26589 fusgrfisbase 26675 |
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