Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vdiscusgrb | Structured version Visualization version GIF version |
Description: A finite simple graph with n vertices is complete iff every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 22-Dec-2020.) |
Ref | Expression |
---|---|
hashnbusgrvd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
vdiscusgrb | ⊢ (𝐺 ∈ FinUSGraph → (𝐺 ∈ ComplUSGraph ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fusgrusgr 27031 | . . 3 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) | |
2 | hashnbusgrvd.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | cusgruvtxb 27131 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
4 | 2 | uvtxssvtx 27099 | . . . . . 6 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉 |
5 | sssseq 3982 | . . . . . . 7 ⊢ ((UnivVtx‘𝐺) ⊆ 𝑉 → (𝑉 ⊆ (UnivVtx‘𝐺) ↔ 𝑉 = (UnivVtx‘𝐺))) | |
6 | eqcom 2825 | . . . . . . 7 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ 𝑉 = (UnivVtx‘𝐺)) | |
7 | 5, 6 | syl6rbbr 291 | . . . . . 6 ⊢ ((UnivVtx‘𝐺) ⊆ 𝑉 → ((UnivVtx‘𝐺) = 𝑉 ↔ 𝑉 ⊆ (UnivVtx‘𝐺))) |
8 | 4, 7 | mp1i 13 | . . . . 5 ⊢ (𝐺 ∈ USGraph → ((UnivVtx‘𝐺) = 𝑉 ↔ 𝑉 ⊆ (UnivVtx‘𝐺))) |
9 | 3, 8 | bitrd 280 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ 𝑉 ⊆ (UnivVtx‘𝐺))) |
10 | dfss3 3953 | . . . 4 ⊢ (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) | |
11 | 9, 10 | syl6bb 288 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
12 | 1, 11 | syl 17 | . 2 ⊢ (𝐺 ∈ FinUSGraph → (𝐺 ∈ ComplUSGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
13 | 2 | usgruvtxvdb 27238 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1))) |
14 | 13 | ralbidva 3193 | . 2 ⊢ (𝐺 ∈ FinUSGraph → (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1))) |
15 | 12, 14 | bitrd 280 | 1 ⊢ (𝐺 ∈ FinUSGraph → (𝐺 ∈ ComplUSGraph ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 1c1 10526 − cmin 10858 ♯chash 13678 Vtxcvtx 26708 USGraphcusgr 26861 FinUSGraphcfusgr 27025 UnivVtxcuvtx 27094 ComplUSGraphccusgr 27119 VtxDegcvtxdg 27174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12881 df-hash 13679 df-edg 26760 df-uhgr 26770 df-ushgr 26771 df-upgr 26794 df-umgr 26795 df-uspgr 26862 df-usgr 26863 df-fusgr 27026 df-nbgr 27042 df-uvtx 27095 df-cplgr 27120 df-cusgr 27121 df-vtxdg 27175 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |