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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 11909 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 12998 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3)) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ 0 ∈ (0...3) |
4 | 1nn0 11907 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 1le3 11843 | . . . . . . 7 ⊢ 1 ≤ 3 | |
6 | elfz2nn0 12992 | . . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
7 | 4, 1, 5, 6 | mpbir3an 1337 | . . . . . 6 ⊢ 1 ∈ (0...3) |
8 | 0ne1 11702 | . . . . . 6 ⊢ 0 ≠ 1 | |
9 | 3, 7, 8 | umgrbi 26880 | . . . . 5 ⊢ {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
11 | 2nn0 11908 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
12 | 2re 11705 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
13 | 3re 11711 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
14 | 2lt3 11803 | . . . . . . . 8 ⊢ 2 < 3 | |
15 | 12, 13, 14 | ltleii 10757 | . . . . . . 7 ⊢ 2 ≤ 3 |
16 | elfz2nn0 12992 | . . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3)) | |
17 | 11, 1, 15, 16 | mpbir3an 1337 | . . . . . 6 ⊢ 2 ∈ (0...3) |
18 | 0ne2 11838 | . . . . . 6 ⊢ 0 ≠ 2 | |
19 | 3, 17, 18 | umgrbi 26880 | . . . . 5 ⊢ {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
21 | nn0fz0 12999 | . . . . . . 7 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3)) | |
22 | 1, 21 | mpbi 232 | . . . . . 6 ⊢ 3 ∈ (0...3) |
23 | 3ne0 11737 | . . . . . . 7 ⊢ 3 ≠ 0 | |
24 | 23 | necomi 3070 | . . . . . 6 ⊢ 0 ≠ 3 |
25 | 3, 22, 24 | umgrbi 26880 | . . . . 5 ⊢ {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
27 | 1ne2 11839 | . . . . . 6 ⊢ 1 ≠ 2 | |
28 | 7, 17, 27 | umgrbi 26880 | . . . . 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 10747 | . . . . . 6 ⊢ 2 ≠ 3 |
31 | 17, 22, 30 | umgrbi 26880 | . . . . 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 14232 | . . 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 1540 | . 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 4542 | . . . 4 ⊢ 𝒫 𝑉 = 𝒫 (0...3) |
38 | 37 | rabeqi 3482 | . . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
39 | 38 | wrdeqi 13881 | . 2 ⊢ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = Word {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
40 | 34, 35, 39 | 3eltr4i 2926 | 1 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 {crab 3142 𝒫 cpw 4538 {cpr 4562 〈cop 4566 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 ≤ cle 10670 2c2 11686 3c3 11687 ℕ0cn0 11891 ...cfz 12886 ♯chash 13684 Word cword 13855 〈“cs7 14202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-s4 14206 df-s5 14207 df-s6 14208 df-s7 14209 |
This theorem is referenced by: konigsbergssiedgwpr 28022 konigsbergumgr 28024 |
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