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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 11918 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 13007 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3)) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ 0 ∈ (0...3) |
4 | 1nn0 11916 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 1le3 11852 | . . . . . . 7 ⊢ 1 ≤ 3 | |
6 | elfz2nn0 13001 | . . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
7 | 4, 1, 5, 6 | mpbir3an 1337 | . . . . . 6 ⊢ 1 ∈ (0...3) |
8 | 0ne1 11711 | . . . . . 6 ⊢ 0 ≠ 1 | |
9 | 3, 7, 8 | umgrbi 26889 | . . . . 5 ⊢ {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
11 | 2nn0 11917 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
12 | 2re 11714 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
13 | 3re 11720 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
14 | 2lt3 11812 | . . . . . . . 8 ⊢ 2 < 3 | |
15 | 12, 13, 14 | ltleii 10766 | . . . . . . 7 ⊢ 2 ≤ 3 |
16 | elfz2nn0 13001 | . . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3)) | |
17 | 11, 1, 15, 16 | mpbir3an 1337 | . . . . . 6 ⊢ 2 ∈ (0...3) |
18 | 0ne2 11847 | . . . . . 6 ⊢ 0 ≠ 2 | |
19 | 3, 17, 18 | umgrbi 26889 | . . . . 5 ⊢ {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
21 | nn0fz0 13008 | . . . . . . 7 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3)) | |
22 | 1, 21 | mpbi 232 | . . . . . 6 ⊢ 3 ∈ (0...3) |
23 | 3ne0 11746 | . . . . . . 7 ⊢ 3 ≠ 0 | |
24 | 23 | necomi 3073 | . . . . . 6 ⊢ 0 ≠ 3 |
25 | 3, 22, 24 | umgrbi 26889 | . . . . 5 ⊢ {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
26 | 25 | a1i 11 | . . . 4 ⊢ (⊤ → {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2}) |
27 | 1ne2 11848 | . . . . . 6 ⊢ 1 ≠ 2 | |
28 | 7, 17, 27 | umgrbi 26889 | . . . . 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 10756 | . . . . . 6 ⊢ 2 ≠ 3 |
31 | 17, 22, 30 | umgrbi 26889 | . . . . 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 14241 | . . 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 1543 | . 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 4560 | . . . 4 ⊢ 𝒫 𝑉 = 𝒫 (0...3) |
38 | 37 | rabeqi 3485 | . . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
39 | 38 | wrdeqi 13890 | . 2 ⊢ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = Word {𝑥 ∈ 𝒫 (0...3) ∣ (♯‘𝑥) = 2} |
40 | 34, 35, 39 | 3eltr4i 2929 | 1 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 {crab 3145 𝒫 cpw 4542 {cpr 4572 〈cop 4576 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 ≤ cle 10679 2c2 11695 3c3 11696 ℕ0cn0 11900 ...cfz 12895 ♯chash 13693 Word cword 13864 〈“cs7 14211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13926 df-s1 13953 df-s2 14213 df-s3 14214 df-s4 14215 df-s5 14216 df-s6 14217 df-s7 14218 |
This theorem is referenced by: konigsbergssiedgwpr 28031 konigsbergumgr 28033 |
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