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Mirrors > Home > MPE Home > Th. List > cusconngr | Structured version Visualization version GIF version |
Description: A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
Ref | Expression |
---|---|
cusconngr | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ ConnGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2733 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | iscplgredg 28654 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑒 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑒)) |
4 | simp-4l 782 | . . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) ∧ 𝑒 ∈ (Edg‘𝐺)) ∧ {𝑘, 𝑛} ⊆ 𝑒) → 𝐺 ∈ UHGraph) | |
5 | simpr 486 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) → 𝑘 ∈ (Vtx‘𝐺)) | |
6 | eldifi 4125 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘}) → 𝑛 ∈ (Vtx‘𝐺)) | |
7 | 5, 6 | anim12i 614 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) → (𝑘 ∈ (Vtx‘𝐺) ∧ 𝑛 ∈ (Vtx‘𝐺))) |
8 | 7 | adantr 482 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑘 ∈ (Vtx‘𝐺) ∧ 𝑛 ∈ (Vtx‘𝐺))) |
9 | 8 | adantr 482 | . . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) ∧ 𝑒 ∈ (Edg‘𝐺)) ∧ {𝑘, 𝑛} ⊆ 𝑒) → (𝑘 ∈ (Vtx‘𝐺) ∧ 𝑛 ∈ (Vtx‘𝐺))) |
10 | id 22 | . . . . . . . . . . 11 ⊢ (𝑒 ∈ (Edg‘𝐺) → 𝑒 ∈ (Edg‘𝐺)) | |
11 | sseq2 4007 | . . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → ({𝑘, 𝑛} ⊆ 𝑐 ↔ {𝑘, 𝑛} ⊆ 𝑒)) | |
12 | 11 | adantl 483 | . . . . . . . . . . 11 ⊢ ((𝑒 ∈ (Edg‘𝐺) ∧ 𝑐 = 𝑒) → ({𝑘, 𝑛} ⊆ 𝑐 ↔ {𝑘, 𝑛} ⊆ 𝑒)) |
13 | 10, 12 | rspcedv 3605 | . . . . . . . . . 10 ⊢ (𝑒 ∈ (Edg‘𝐺) → ({𝑘, 𝑛} ⊆ 𝑒 → ∃𝑐 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑐)) |
14 | 13 | adantl 483 | . . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) ∧ 𝑒 ∈ (Edg‘𝐺)) → ({𝑘, 𝑛} ⊆ 𝑒 → ∃𝑐 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑐)) |
15 | 14 | imp 408 | . . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) ∧ 𝑒 ∈ (Edg‘𝐺)) ∧ {𝑘, 𝑛} ⊆ 𝑒) → ∃𝑐 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑐) |
16 | 1, 2 | 1pthon2v 29386 | . . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑘 ∈ (Vtx‘𝐺) ∧ 𝑛 ∈ (Vtx‘𝐺)) ∧ ∃𝑐 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑐) → ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) |
17 | 4, 9, 15, 16 | syl3anc 1372 | . . . . . . 7 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) ∧ 𝑒 ∈ (Edg‘𝐺)) ∧ {𝑘, 𝑛} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) |
18 | 17 | rexlimdva2 3158 | . . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})) → (∃𝑒 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
19 | 18 | ralimdva 3168 | . . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑘 ∈ (Vtx‘𝐺)) → (∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑒 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑒 → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
20 | 19 | ralimdva 3168 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑒 ∈ (Edg‘𝐺){𝑘, 𝑛} ⊆ 𝑒 → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
21 | 3, 20 | sylbid 239 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐺 ∈ ComplGraph → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
22 | 21 | imp 408 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐺 ∈ ComplGraph) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝) |
23 | 1 | isconngr1 29423 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
24 | 23 | adantr 482 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐺 ∈ ComplGraph) → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
25 | 22, 24 | mpbird 257 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ ConnGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 {cpr 4629 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 Vtxcvtx 28236 Edgcedg 28287 UHGraphcuhgr 28296 ComplGraphccplgr 28646 PathsOncpthson 28951 ConnGraphcconngr 29419 |
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 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-edg 28288 df-uhgr 28298 df-nbgr 28570 df-uvtx 28623 df-cplgr 28648 df-wlks 28836 df-wlkson 28837 df-trls 28929 df-trlson 28930 df-pths 28953 df-pthson 28955 df-conngr 29420 |
This theorem is referenced by: (None) |
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