![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 47843 | . 2 ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} |
4 | usgrexmpl2.g | . . . 4 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
5 | 4 | eleq1i 2835 | . . 3 ⊢ (𝐺 ∈ USGraph ↔ 〈𝑉, 𝐸〉 ∈ USGraph) |
6 | 1 | ovexi 7484 | . . . 4 ⊢ 𝑉 ∈ V |
7 | s7cli 14936 | . . . . 5 ⊢ 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 ∈ Word V | |
8 | 2, 7 | eqeltri 2840 | . . . 4 ⊢ 𝐸 ∈ Word V |
9 | isusgrop 29199 | . . . 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 2108 {crab 3443 Vcvv 3488 𝒫 cpw 4622 {cpr 4650 〈cop 4654 dom cdm 5700 –1-1→wf1 6572 ‘cfv 6575 (class class class)co 7450 0cc0 11186 1c1 11187 2c2 12350 3c3 12351 4c4 12352 5c5 12353 ...cfz 13569 ♯chash 14381 Word cword 14564 〈“cs7 14897 USGraphcusgr 29186 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-oadd 8528 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-dju 9972 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-n0 12556 df-xnn0 12628 df-z 12642 df-uz 12906 df-fz 13570 df-fzo 13714 df-hash 14382 df-word 14565 df-concat 14621 df-s1 14646 df-s2 14899 df-s3 14900 df-s4 14901 df-s5 14902 df-s6 14903 df-s7 14904 df-vtx 29035 df-iedg 29036 df-usgr 29188 |
This theorem is referenced by: usgrexmpl2nblem 47847 usgrexmpl2trifr 47854 usgrexmpl12ngric 47855 usgrexmpl12ngrlic 47856 |
Copyright terms: Public domain | W3C validator |