Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > uvtxnm1nbgr | Structured version Visualization version GIF version |
Description: A universal vertex has 𝑛 − 1 neighbors in a finite graph with 𝑛 vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.) |
Ref | Expression |
---|---|
uvtxnm1nbgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
uvtxnm1nbgr | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (♯‘(𝐺 NeighbVtx 𝑁)) = ((♯‘𝑉) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxnm1nbgr.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | uvtxnbgr 27182 | . . . 4 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
3 | 2 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
4 | 3 | fveq2d 6674 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘(𝑉 ∖ {𝑁}))) |
5 | 1 | fusgrvtxfi 27101 | . . 3 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
6 | 1 | uvtxisvtx 27171 | . . . 4 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ 𝑉) |
7 | 6 | snssd 4742 | . . 3 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → {𝑁} ⊆ 𝑉) |
8 | hashssdif 13774 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ {𝑁} ⊆ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − (♯‘{𝑁}))) | |
9 | 5, 7, 8 | syl2an 597 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − (♯‘{𝑁}))) |
10 | hashsng 13731 | . . . 4 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (♯‘{𝑁}) = 1) | |
11 | 10 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (♯‘{𝑁}) = 1) |
12 | 11 | oveq2d 7172 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → ((♯‘𝑉) − (♯‘{𝑁})) = ((♯‘𝑉) − 1)) |
13 | 4, 9, 12 | 3eqtrd 2860 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (♯‘(𝐺 NeighbVtx 𝑁)) = ((♯‘𝑉) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 1c1 10538 − cmin 10870 ♯chash 13691 Vtxcvtx 26781 FinUSGraphcfusgr 27098 NeighbVtx cnbgr 27114 UnivVtxcuvtx 27167 |
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-fal 1550 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-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-fusgr 27099 df-nbgr 27115 df-uvtx 27168 |
This theorem is referenced by: uvtxnbvtxm1 27188 |
Copyright terms: Public domain | W3C validator |