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Mirrors > Home > MPE Home > Th. List > konigsbergiedgw | Structured version Visualization version GIF version |
Description: The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-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 |
---|---|
konigsbergiedgw | ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 12544 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 13654 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3)) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ 0 ∈ (0...3) |
4 | 1nn0 12542 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 1le3 12478 | . . . . . . 7 ⊢ 1 ≤ 3 | |
6 | elfz2nn0 13648 | . . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
7 | 4, 1, 5, 6 | mpbir3an 1338 | . . . . . 6 ⊢ 1 ∈ (0...3) |
8 | 0ne1 12337 | . . . . . 6 ⊢ 0 ≠ 1 | |
9 | 3, 7, 8 | umgrbi 29040 | . . . . 5 ⊢ {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
11 | 2nn0 12543 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
12 | 2re 12340 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
13 | 3re 12346 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
14 | 2lt3 12438 | . . . . . . . 8 ⊢ 2 < 3 | |
15 | 12, 13, 14 | ltleii 11389 | . . . . . . 7 ⊢ 2 ≤ 3 |
16 | elfz2nn0 13648 | . . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3)) | |
17 | 11, 1, 15, 16 | mpbir3an 1338 | . . . . . 6 ⊢ 2 ∈ (0...3) |
18 | 0ne2 12473 | . . . . . 6 ⊢ 0 ≠ 2 | |
19 | 3, 17, 18 | umgrbi 29040 | . . . . 5 ⊢ {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
21 | nn0fz0 13655 | . . . . . . 7 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3)) | |
22 | 1, 21 | mpbi 229 | . . . . . 6 ⊢ 3 ∈ (0...3) |
23 | 3ne0 12372 | . . . . . . 7 ⊢ 3 ≠ 0 | |
24 | 23 | necomi 2985 | . . . . . 6 ⊢ 0 ≠ 3 |
25 | 3, 22, 24 | umgrbi 29040 | . . . . 5 ⊢ {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
27 | 1ne2 12474 | . . . . . 6 ⊢ 1 ≠ 2 | |
28 | 7, 17, 27 | umgrbi 29040 | . . . . 5 ⊢ {1, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
29 | 28 | a1i 11 | . . . 4 ⊢ (⊤ → {1, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
30 | 12, 14 | ltneii 11379 | . . . . . 6 ⊢ 2 ≠ 3 |
31 | 17, 22, 30 | umgrbi 29040 | . . . . 5 ⊢ {2, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
32 | 31 | a1i 11 | . . . 4 ⊢ (⊤ → {2, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
33 | 10, 20, 26, 29, 29, 32, 32 | s7cld 14887 | . . 3 ⊢ (⊤ → 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
34 | 33 | mptru 1541 | . 2 ⊢ 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
35 | konigsberg.e | . 2 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
36 | konigsberg.v | . . . . 5 ⊢ 𝑉 = (0...3) | |
37 | 36 | pweqi 4623 | . . . 4 ⊢ 𝒫 𝑉 = 𝒫 (0...3) |
38 | 37 | rabeqi 3433 | . . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
39 | 38 | wrdeqi 14547 | . 2 ⊢ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = Word {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
40 | 34, 35, 39 | 3eltr4i 2839 | 1 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 {crab 3419 𝒫 cpw 4607 {cpr 4635 〈cop 4639 class class class wbr 5155 ‘cfv 6556 (class class class)co 7426 0cc0 11160 1c1 11161 ≤ cle 11301 2c2 12321 3c3 12322 ℕ0cn0 12526 ...cfz 13540 ♯chash 14349 Word cword 14524 〈“cs7 14857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-oadd 8502 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-dju 9946 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12613 df-uz 12877 df-fz 13541 df-fzo 13684 df-hash 14350 df-word 14525 df-concat 14581 df-s1 14606 df-s2 14859 df-s3 14860 df-s4 14861 df-s5 14862 df-s6 14863 df-s7 14864 |
This theorem is referenced by: konigsbergssiedgwpr 30185 konigsbergumgr 30187 |
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