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Mirrors > Home > MPE Home > Th. List > Mathboxes > usgrexmpl2 | Structured version Visualization version GIF version |
Description: 𝐺 is a simple graph of six vertices 0, 1, 2, 3, 4, 5, with edges {0, 1}, {1, 2}, {2, 3}, {0, 3}, {3, 4}, {4, 5}, {0, 5}. (Contributed by AV, 3-Aug-2025.) |
Ref | Expression |
---|---|
usgrexmpl2.v | ⊢ 𝑉 = (0...5) |
usgrexmpl2.e | ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 |
usgrexmpl2.g | ⊢ 𝐺 = 〈𝑉, 𝐸〉 |
Ref | Expression |
---|---|
usgrexmpl2 | ⊢ 𝐺 ∈ USGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrexmpl2.v | . . 3 ⊢ 𝑉 = (0...5) | |
2 | usgrexmpl2.e | . . 3 ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 | |
3 | 1, 2 | usgrexmpl2lem 47761 | . 2 ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} |
4 | usgrexmpl2.g | . . . 4 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
5 | 4 | eleq1i 2829 | . . 3 ⊢ (𝐺 ∈ USGraph ↔ 〈𝑉, 𝐸〉 ∈ USGraph) |
6 | 1 | ovexi 7479 | . . . 4 ⊢ 𝑉 ∈ V |
7 | s7cli 14930 | . . . . 5 ⊢ 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 ∈ Word V | |
8 | 2, 7 | eqeltri 2834 | . . . 4 ⊢ 𝐸 ∈ Word V |
9 | isusgrop 29188 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (〈𝑉, 𝐸〉 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})) | |
10 | 6, 8, 9 | mp2an 691 | . . 3 ⊢ (〈𝑉, 𝐸〉 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
11 | 5, 10 | bitri 275 | . 2 ⊢ (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
12 | 3, 11 | mpbir 231 | 1 ⊢ 𝐺 ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2103 {crab 3438 Vcvv 3482 𝒫 cpw 4622 {cpr 4650 〈cop 4654 dom cdm 5699 –1-1→wf1 6569 ‘cfv 6572 (class class class)co 7445 0cc0 11180 1c1 11181 2c2 12344 3c3 12345 4c4 12346 5c5 12347 ...cfz 13563 ♯chash 14375 Word cword 14558 〈“cs7 14891 USGraphcusgr 29175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-oadd 8522 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-dju 9966 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-n0 12550 df-xnn0 12622 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-hash 14376 df-word 14559 df-concat 14615 df-s1 14640 df-s2 14893 df-s3 14894 df-s4 14895 df-s5 14896 df-s6 14897 df-s7 14898 df-vtx 29024 df-iedg 29025 df-usgr 29177 |
This theorem is referenced by: usgrexmpl2nblem 47765 usgrexmpl2trifr 47772 usgrexmpl12ngric 47773 |
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