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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 2730 | . . . . 5 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
3 | uhgr0vb 28977 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) | |
4 | 3 | biimpd 228 | . . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)) |
5 | 4 | ex 411 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))) |
6 | 2, 5 | biimtrid 241 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))) |
7 | 6 | pm2.43a 54 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅)) |
8 | 7 | imp 405 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅) |
9 | uhgr0v0e.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
10 | 9 | eqeq1i 2730 | . . . 4 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅) |
11 | uhgriedg0edg0 29032 | . . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
12 | 10, 11 | bitrid 282 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
13 | 12 | adantr 479 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
14 | 8, 13 | mpbird 256 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∅c0 4322 ‘cfv 6549 Vtxcvtx 28901 iEdgciedg 28902 Edgcedg 28952 UHGraphcuhgr 28961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-edg 28953 df-uhgr 28963 |
This theorem is referenced by: uhgr0vsize0 29144 uhgr0vusgr 29147 fusgrfisbase 29233 |
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