![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uhgrvd00 | Structured version Visualization version GIF version |
Description: If every vertex in a hypergraph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.) |
Ref | Expression |
---|---|
vtxdusgradjvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdusgradjvtx.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
uhgrvd00 | ⊢ (𝐺 ∈ UHGraph → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdusgradjvtx.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | vtxdusgradjvtx.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | eqid 2825 | . . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
4 | 1, 2, 3 | vtxduhgr0edgnel 26799 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑣 ∈ 𝑒)) |
5 | ralnex 3201 | . . . 4 ⊢ (∀𝑒 ∈ 𝐸 ¬ 𝑣 ∈ 𝑒 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑣 ∈ 𝑒) | |
6 | 4, 5 | syl6bbr 281 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 0 ↔ ∀𝑒 ∈ 𝐸 ¬ 𝑣 ∈ 𝑒)) |
7 | 6 | ralbidva 3194 | . 2 ⊢ (𝐺 ∈ UHGraph → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 ↔ ∀𝑣 ∈ 𝑉 ∀𝑒 ∈ 𝐸 ¬ 𝑣 ∈ 𝑒)) |
8 | ralcom 3308 | . . . . 5 ⊢ (∀𝑣 ∈ 𝑉 ∀𝑒 ∈ 𝐸 ¬ 𝑣 ∈ 𝑒 ↔ ∀𝑒 ∈ 𝐸 ∀𝑣 ∈ 𝑉 ¬ 𝑣 ∈ 𝑒) | |
9 | ralnex2 3255 | . . . . 5 ⊢ (∀𝑒 ∈ 𝐸 ∀𝑣 ∈ 𝑉 ¬ 𝑣 ∈ 𝑒 ↔ ¬ ∃𝑒 ∈ 𝐸 ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒) | |
10 | 8, 9 | bitri 267 | . . . 4 ⊢ (∀𝑣 ∈ 𝑉 ∀𝑒 ∈ 𝐸 ¬ 𝑣 ∈ 𝑒 ↔ ¬ ∃𝑒 ∈ 𝐸 ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒) |
11 | simpr 479 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐸) | |
12 | 2 | eleq2i 2898 | . . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺)) |
13 | uhgredgn0 26433 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) | |
14 | 12, 13 | sylan2b 587 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
15 | eldifsn 4538 | . . . . . . . . . . 11 ⊢ (𝑒 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅)) | |
16 | elpwi 4390 | . . . . . . . . . . . . 13 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺)) | |
17 | 1 | sseq2i 3855 | . . . . . . . . . . . . . 14 ⊢ (𝑒 ⊆ 𝑉 ↔ 𝑒 ⊆ (Vtx‘𝐺)) |
18 | ssn0rex 4167 | . . . . . . . . . . . . . . 15 ⊢ ((𝑒 ⊆ 𝑉 ∧ 𝑒 ≠ ∅) → ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒) | |
19 | 18 | ex 403 | . . . . . . . . . . . . . 14 ⊢ (𝑒 ⊆ 𝑉 → (𝑒 ≠ ∅ → ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒)) |
20 | 17, 19 | sylbir 227 | . . . . . . . . . . . . 13 ⊢ (𝑒 ⊆ (Vtx‘𝐺) → (𝑒 ≠ ∅ → ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒)) |
21 | 16, 20 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → (𝑒 ≠ ∅ → ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒)) |
22 | 21 | imp 397 | . . . . . . . . . . 11 ⊢ ((𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅) → ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒) |
23 | 15, 22 | sylbi 209 | . . . . . . . . . 10 ⊢ (𝑒 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒) |
24 | 14, 23 | syl 17 | . . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒) |
25 | 11, 24 | jca 507 | . . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → (𝑒 ∈ 𝐸 ∧ ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒)) |
26 | 25 | ex 403 | . . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (𝑒 ∈ 𝐸 → (𝑒 ∈ 𝐸 ∧ ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒))) |
27 | 26 | eximdv 2016 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 𝑒 ∈ 𝐸 → ∃𝑒(𝑒 ∈ 𝐸 ∧ ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒))) |
28 | n0 4162 | . . . . . 6 ⊢ (𝐸 ≠ ∅ ↔ ∃𝑒 𝑒 ∈ 𝐸) | |
29 | df-rex 3123 | . . . . . 6 ⊢ (∃𝑒 ∈ 𝐸 ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒 ↔ ∃𝑒(𝑒 ∈ 𝐸 ∧ ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒)) | |
30 | 27, 28, 29 | 3imtr4g 288 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝐸 ≠ ∅ → ∃𝑒 ∈ 𝐸 ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒)) |
31 | 30 | con3d 150 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (¬ ∃𝑒 ∈ 𝐸 ∃𝑣 ∈ 𝑉 𝑣 ∈ 𝑒 → ¬ 𝐸 ≠ ∅)) |
32 | 10, 31 | syl5bi 234 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∀𝑣 ∈ 𝑉 ∀𝑒 ∈ 𝐸 ¬ 𝑣 ∈ 𝑒 → ¬ 𝐸 ≠ ∅)) |
33 | nne 3003 | . . 3 ⊢ (¬ 𝐸 ≠ ∅ ↔ 𝐸 = ∅) | |
34 | 32, 33 | syl6ib 243 | . 2 ⊢ (𝐺 ∈ UHGraph → (∀𝑣 ∈ 𝑉 ∀𝑒 ∈ 𝐸 ¬ 𝑣 ∈ 𝑒 → 𝐸 = ∅)) |
35 | 7, 34 | sylbid 232 | 1 ⊢ (𝐺 ∈ UHGraph → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∃wex 1878 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 ∖ cdif 3795 ⊆ wss 3798 ∅c0 4146 𝒫 cpw 4380 {csn 4399 ‘cfv 6127 0cc0 10259 Vtxcvtx 26301 Edgcedg 26352 UHGraphcuhgr 26361 VtxDegcvtxdg 26770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-xnn0 11698 df-z 11712 df-uz 11976 df-xadd 12240 df-fz 12627 df-hash 13418 df-edg 26353 df-uhgr 26363 df-vtxdg 26771 |
This theorem is referenced by: usgrvd00 26840 uhgr0edg0rgrb 26879 |
Copyright terms: Public domain | W3C validator |