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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 11768 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 12859 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3)) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ 0 ∈ (0...3) |
4 | 1nn0 11766 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 1le3 11702 | . . . . . . 7 ⊢ 1 ≤ 3 | |
6 | elfz2nn0 12853 | . . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
7 | 4, 1, 5, 6 | mpbir3an 1334 | . . . . . 6 ⊢ 1 ∈ (0...3) |
8 | 0ne1 11561 | . . . . . 6 ⊢ 0 ≠ 1 | |
9 | 3, 7, 8 | umgrbi 26574 | . . . . 5 ⊢ {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
11 | 2nn0 11767 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
12 | 2re 11564 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
13 | 3re 11570 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
14 | 2lt3 11662 | . . . . . . . 8 ⊢ 2 < 3 | |
15 | 12, 13, 14 | ltleii 10615 | . . . . . . 7 ⊢ 2 ≤ 3 |
16 | elfz2nn0 12853 | . . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3)) | |
17 | 11, 1, 15, 16 | mpbir3an 1334 | . . . . . 6 ⊢ 2 ∈ (0...3) |
18 | 0ne2 11697 | . . . . . 6 ⊢ 0 ≠ 2 | |
19 | 3, 17, 18 | umgrbi 26574 | . . . . 5 ⊢ {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
21 | nn0fz0 12860 | . . . . . . 7 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3)) | |
22 | 1, 21 | mpbi 231 | . . . . . 6 ⊢ 3 ∈ (0...3) |
23 | 3ne0 11596 | . . . . . . 7 ⊢ 3 ≠ 0 | |
24 | 23 | necomi 3038 | . . . . . 6 ⊢ 0 ≠ 3 |
25 | 3, 22, 24 | umgrbi 26574 | . . . . 5 ⊢ {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
27 | 1ne2 11698 | . . . . . 6 ⊢ 1 ≠ 2 | |
28 | 7, 17, 27 | umgrbi 26574 | . . . . 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 10605 | . . . . . 6 ⊢ 2 ≠ 3 |
31 | 17, 22, 30 | umgrbi 26574 | . . . . 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 14079 | . . 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 1529 | . 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 4461 | . . . 4 ⊢ 𝒫 𝑉 = 𝒫 (0...3) |
38 | 37 | rabeqi 3427 | . . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
39 | 38 | wrdeqi 13738 | . 2 ⊢ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = Word {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
40 | 34, 35, 39 | 3eltr4i 2896 | 1 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ⊤wtru 1523 ∈ wcel 2081 {crab 3109 𝒫 cpw 4457 {cpr 4478 〈cop 4482 class class class wbr 4966 ‘cfv 6230 (class class class)co 7021 0cc0 10388 1c1 10389 ≤ cle 10527 2c2 11545 3c3 11546 ℕ0cn0 11750 ...cfz 12747 ♯chash 13545 Word cword 13712 〈“cs7 14049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-dju 9181 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-fzo 12889 df-hash 13546 df-word 13713 df-concat 13774 df-s1 13799 df-s2 14051 df-s3 14052 df-s4 14053 df-s5 14054 df-s6 14055 df-s7 14056 |
This theorem is referenced by: konigsbergssiedgwpr 27723 konigsbergumgr 27725 |
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