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Mirrors > Home > MPE Home > Th. List > fusgrn0degnn0 | Structured version Visualization version GIF version |
Description: In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 1-Apr-2021.) |
Ref | Expression |
---|---|
fusgrn0degnn0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
fusgrn0degnn0 | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4307 | . . 3 ⊢ (𝑉 ≠ ∅ ↔ ∃𝑘 𝑘 ∈ 𝑉) | |
2 | fusgrn0degnn0.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | vtxdgfusgr 27207 | . . . . 5 ⊢ (𝐺 ∈ FinUSGraph → ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ0) |
4 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑢 = 𝑘 → ((VtxDeg‘𝐺)‘𝑢) = ((VtxDeg‘𝐺)‘𝑘)) | |
5 | 4 | eleq1d 2894 | . . . . . . 7 ⊢ (𝑢 = 𝑘 → (((VtxDeg‘𝐺)‘𝑢) ∈ ℕ0 ↔ ((VtxDeg‘𝐺)‘𝑘) ∈ ℕ0)) |
6 | 5 | rspcv 3615 | . . . . . 6 ⊢ (𝑘 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ0 → ((VtxDeg‘𝐺)‘𝑘) ∈ ℕ0)) |
7 | risset 3264 | . . . . . . . 8 ⊢ (((VtxDeg‘𝐺)‘𝑘) ∈ ℕ0 ↔ ∃𝑛 ∈ ℕ0 𝑛 = ((VtxDeg‘𝐺)‘𝑘)) | |
8 | fveqeq2 6672 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑘 → (((VtxDeg‘𝐺)‘𝑣) = 𝑛 ↔ ((VtxDeg‘𝐺)‘𝑘) = 𝑛)) | |
9 | eqcom 2825 | . . . . . . . . . . . 12 ⊢ (((VtxDeg‘𝐺)‘𝑘) = 𝑛 ↔ 𝑛 = ((VtxDeg‘𝐺)‘𝑘)) | |
10 | 8, 9 | syl6bb 288 | . . . . . . . . . . 11 ⊢ (𝑣 = 𝑘 → (((VtxDeg‘𝐺)‘𝑣) = 𝑛 ↔ 𝑛 = ((VtxDeg‘𝐺)‘𝑘))) |
11 | 10 | rexbidv 3294 | . . . . . . . . . 10 ⊢ (𝑣 = 𝑘 → (∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛 ↔ ∃𝑛 ∈ ℕ0 𝑛 = ((VtxDeg‘𝐺)‘𝑘))) |
12 | 11 | rspcev 3620 | . . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑉 ∧ ∃𝑛 ∈ ℕ0 𝑛 = ((VtxDeg‘𝐺)‘𝑘)) → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛) |
13 | 12 | expcom 414 | . . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ0 𝑛 = ((VtxDeg‘𝐺)‘𝑘) → (𝑘 ∈ 𝑉 → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)) |
14 | 7, 13 | sylbi 218 | . . . . . . 7 ⊢ (((VtxDeg‘𝐺)‘𝑘) ∈ ℕ0 → (𝑘 ∈ 𝑉 → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)) |
15 | 14 | com12 32 | . . . . . 6 ⊢ (𝑘 ∈ 𝑉 → (((VtxDeg‘𝐺)‘𝑘) ∈ ℕ0 → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)) |
16 | 6, 15 | syld 47 | . . . . 5 ⊢ (𝑘 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ0 → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)) |
17 | 3, 16 | syl5 34 | . . . 4 ⊢ (𝑘 ∈ 𝑉 → (𝐺 ∈ FinUSGraph → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)) |
18 | 17 | exlimiv 1922 | . . 3 ⊢ (∃𝑘 𝑘 ∈ 𝑉 → (𝐺 ∈ FinUSGraph → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)) |
19 | 1, 18 | sylbi 218 | . 2 ⊢ (𝑉 ≠ ∅ → (𝐺 ∈ FinUSGraph → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)) |
20 | 19 | impcom 408 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ∅c0 4288 ‘cfv 6348 ℕ0cn0 11885 Vtxcvtx 26708 FinUSGraphcfusgr 27025 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-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-vtx 26710 df-iedg 26711 df-edg 26760 df-uhgr 26770 df-upgr 26794 df-umgr 26795 df-uspgr 26862 df-usgr 26863 df-fusgr 27026 df-vtxdg 27175 |
This theorem is referenced by: friendshipgt3 28104 |
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