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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 27037 | . . . . . . . 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 26968 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
13 | uhgriedg0edg0 26912 | . . . . . . . . 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 1934 | . . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑣 𝑉 = {𝑣} → 𝐸 = ∅)) |
20 | 3 | fvexi 6684 | . . . 4 ⊢ 𝑉 ∈ V |
21 | hash1snb 13781 | . . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) | |
22 | 20, 21 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) |
23 | 10 | fvexi 6684 | . . . 4 ⊢ 𝐸 ∈ V |
24 | hasheq0 13725 | . . . 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 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 {csn 4567 ‘cfv 6355 0cc0 10537 1c1 10538 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 Edgcedg 26832 UHGraphcuhgr 26841 USGraphcusgr 26934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-uspgr 26935 df-usgr 26936 |
This theorem is referenced by: cusgrsizeindb1 27232 |
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