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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 485 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐺 ∈ USGraph) | |
2 | vex 3497 | . . . . . . . 8 ⊢ 𝑣 ∈ V | |
3 | fusgredgfi.v | . . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | eqeq1i 2826 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑣} ↔ (Vtx‘𝐺) = {𝑣}) |
5 | 4 | biimpi 218 | . . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (Vtx‘𝐺) = {𝑣}) |
6 | 5 | adantl 484 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (Vtx‘𝐺) = {𝑣}) |
7 | usgr1vr 27036 | . . . . . . . 8 ⊢ ((𝑣 ∈ V ∧ (Vtx‘𝐺) = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) | |
8 | 2, 6, 7 | sylancr 589 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
9 | 1, 8 | mpd 15 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (iEdg‘𝐺) = ∅) |
10 | fusgredgfi.e | . . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺) | |
11 | 10 | eqeq1i 2826 | . . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅) |
12 | usgruhgr 26967 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
13 | uhgriedg0edg0 26911 | . . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
14 | 12, 13 | syl 17 | . . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
15 | 14 | adantr 483 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
16 | 11, 15 | syl5bb 285 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
17 | 9, 16 | mpbird 259 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐸 = ∅) |
18 | 17 | ex 415 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝑉 = {𝑣} → 𝐸 = ∅)) |
19 | 18 | exlimdv 1930 | . . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑣 𝑉 = {𝑣} → 𝐸 = ∅)) |
20 | 3 | fvexi 6683 | . . . 4 ⊢ 𝑉 ∈ V |
21 | hash1snb 13779 | . . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) | |
22 | 20, 21 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) |
23 | 10 | fvexi 6683 | . . . 4 ⊢ 𝐸 ∈ V |
24 | hasheq0 13723 | . . . 4 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) | |
25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) |
26 | 19, 22, 25 | 3imtr4d 296 | . 2 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 1 → (♯‘𝐸) = 0)) |
27 | 26 | imp 409 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 {csn 4566 ‘cfv 6354 0cc0 10536 1c1 10537 ♯chash 13689 Vtxcvtx 26780 iEdgciedg 26781 Edgcedg 26831 UHGraphcuhgr 26840 USGraphcusgr 26933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-hash 13690 df-edg 26832 df-uhgr 26842 df-upgr 26866 df-uspgr 26934 df-usgr 26935 |
This theorem is referenced by: cusgrsizeindb1 27231 |
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