![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > konigsbergumgr | Structured version Visualization version GIF version |
Description: The Königsberg graph 𝐺 is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.) |
Ref | Expression |
---|---|
konigsberg.v | ⊢ 𝑉 = (0...3) |
konigsberg.e | ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 |
konigsberg.g | ⊢ 𝐺 = 〈𝑉, 𝐸〉 |
Ref | Expression |
---|---|
konigsbergumgr | ⊢ 𝐺 ∈ UMGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | konigsberg.v | . . 3 ⊢ 𝑉 = (0...3) | |
2 | konigsberg.e | . . 3 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
3 | konigsberg.g | . . 3 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
4 | 1, 2, 3 | konigsbergiedgw 29491 | . 2 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
5 | opex 5464 | . . . 4 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
6 | 3, 5 | eqeltri 2830 | . . 3 ⊢ 𝐺 ∈ V |
7 | s7cli 14833 | . . . 4 ⊢ 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 ∈ Word V | |
8 | 2, 7 | eqeltri 2830 | . . 3 ⊢ 𝐸 ∈ Word V |
9 | 1, 2, 3 | konigsbergvtx 29489 | . . . . 5 ⊢ (Vtx‘𝐺) = (0...3) |
10 | 1, 9 | eqtr4i 2764 | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) |
11 | 1, 2, 3 | konigsbergiedg 29490 | . . . . 5 ⊢ (iEdg‘𝐺) = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 |
12 | 2, 11 | eqtr4i 2764 | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) |
13 | 10, 12 | wrdumgr 28347 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐸 ∈ Word V) → (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) |
14 | 6, 8, 13 | mp2an 691 | . 2 ⊢ (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
15 | 4, 14 | mpbir 230 | 1 ⊢ 𝐺 ∈ UMGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 𝒫 cpw 4602 {cpr 4630 〈cop 4634 ‘cfv 6541 (class class class)co 7406 0cc0 11107 1c1 11108 2c2 12264 3c3 12265 ...cfz 13481 ♯chash 14287 Word cword 14461 〈“cs7 14794 Vtxcvtx 28246 iEdgciedg 28247 UMGraphcumgr 28331 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-hash 14288 df-word 14462 df-concat 14518 df-s1 14543 df-s2 14796 df-s3 14797 df-s4 14798 df-s5 14799 df-s6 14800 df-s7 14801 df-vtx 28248 df-iedg 28249 df-umgr 28333 |
This theorem is referenced by: konigsberg 29500 |
Copyright terms: Public domain | W3C validator |