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Mirrors > Home > MPE Home > Th. List > hashnbusgrnn0 | Structured version Visualization version GIF version |
Description: The number of neighbors of a vertex in a finite simple graph is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 15-Dec-2020.) |
Ref | Expression |
---|---|
hashnbusgrnn0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
hashnbusgrnn0 | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnbusgrnn0.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | eleq2i 2830 | . . 3 ⊢ (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ (Vtx‘𝐺)) |
3 | nbfiusgrfi 28152 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑈) ∈ Fin) | |
4 | 2, 3 | sylan2b 595 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (𝐺 NeighbVtx 𝑈) ∈ Fin) |
5 | hashcl 14210 | . 2 ⊢ ((𝐺 NeighbVtx 𝑈) ∈ Fin → (♯‘(𝐺 NeighbVtx 𝑈)) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6494 (class class class)co 7352 Fincfn 8842 ℕ0cn0 12372 ♯chash 14184 Vtxcvtx 27776 FinUSGraphcfusgr 28093 NeighbVtx cnbgr 28109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-dju 9796 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-n0 12373 df-xnn0 12445 df-z 12459 df-uz 12723 df-fz 13380 df-hash 14185 df-vtx 27778 df-iedg 27779 df-edg 27828 df-uhgr 27838 df-upgr 27862 df-umgr 27863 df-uspgr 27930 df-usgr 27931 df-fusgr 28094 df-nbgr 28110 |
This theorem is referenced by: (None) |
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