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Mirrors > Home > MPE Home > Th. List > usgr1v0e | Structured version Visualization version GIF version |
Description: The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.) |
Ref | Expression |
---|---|
fusgredgfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
fusgredgfi.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
usgr1v0e | ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 476 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐺 ∈ USGraph) | |
2 | vex 3401 | . . . . . . . . 9 ⊢ 𝑣 ∈ V | |
3 | 2 | a1i 11 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝑣 ∈ V) |
4 | fusgredgfi.v | . . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | eqeq1i 2783 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑣} ↔ (Vtx‘𝐺) = {𝑣}) |
6 | 5 | biimpi 208 | . . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (Vtx‘𝐺) = {𝑣}) |
7 | 6 | adantl 475 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (Vtx‘𝐺) = {𝑣}) |
8 | usgr1vr 26602 | . . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ (Vtx‘𝐺) = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) | |
9 | 8 | 3adant1 1121 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ V ∧ (Vtx‘𝐺) = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
10 | 1, 3, 7, 9 | syl3anc 1439 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
11 | 1, 10 | mpd 15 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (iEdg‘𝐺) = ∅) |
12 | fusgredgfi.e | . . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺) | |
13 | 12 | eqeq1i 2783 | . . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅) |
14 | usgruhgr 26532 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
15 | uhgriedg0edg0 26475 | . . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
17 | 16 | adantr 474 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
18 | 13, 17 | syl5bb 275 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
19 | 11, 18 | mpbird 249 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐸 = ∅) |
20 | 19 | ex 403 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝑉 = {𝑣} → 𝐸 = ∅)) |
21 | 20 | exlimdv 1976 | . . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑣 𝑉 = {𝑣} → 𝐸 = ∅)) |
22 | 4 | fvexi 6460 | . . . 4 ⊢ 𝑉 ∈ V |
23 | hash1snb 13521 | . . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) | |
24 | 22, 23 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) |
25 | 12 | fvexi 6460 | . . . 4 ⊢ 𝐸 ∈ V |
26 | hasheq0 13469 | . . . 4 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) | |
27 | 25, 26 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) |
28 | 21, 24, 27 | 3imtr4d 286 | . 2 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 1 → (♯‘𝐸) = 0)) |
29 | 28 | imp 397 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 {csn 4398 ‘cfv 6135 0cc0 10272 1c1 10273 ♯chash 13435 Vtxcvtx 26344 iEdgciedg 26345 Edgcedg 26395 UHGraphcuhgr 26404 USGraphcusgr 26498 |
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 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-hash 13436 df-edg 26396 df-uhgr 26406 df-upgr 26430 df-uspgr 26499 df-usgr 26500 |
This theorem is referenced by: cusgrsizeindb1 26798 |
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