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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 2627 | . . . . 5 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
3 | uhgr0vb 25967 | . . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) | |
4 | 3 | biimpd 219 | . . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)) |
5 | 4 | ex 450 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))) |
6 | 2, 5 | syl5bi 232 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))) |
7 | 6 | pm2.43a 54 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅)) |
8 | 7 | imp 445 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅) |
9 | uhgr0v0e.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
10 | 9 | eqeq1i 2627 | . . . 4 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅) |
11 | uhgriedg0edg0 26022 | . . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
12 | 10, 11 | syl5bb 272 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
13 | 12 | adantr 481 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
14 | 8, 13 | mpbird 247 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 ‘cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 Edgcedg 25939 UHGraph cuhgr 25951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-edg 25940 df-uhgr 25953 |
This theorem is referenced by: uhgr0vsize0 26131 uhgr0vusgr 26134 fusgrfisbase 26220 |
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