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Mirrors > Home > MPE Home > Th. List > konigsberglem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for konigsberg 28030: Vertices 0, 1, 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised 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 |
---|---|
konigsberglem4 | ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 11909 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 12998 | . . . . . 6 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...3) |
4 | konigsberg.v | . . . . 5 ⊢ 𝑉 = (0...3) | |
5 | 3, 4 | eleqtrri 2912 | . . . 4 ⊢ 0 ∈ 𝑉 |
6 | n2dvds3 15715 | . . . . 5 ⊢ ¬ 2 ∥ 3 | |
7 | konigsberg.e | . . . . . . 7 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
8 | konigsberg.g | . . . . . . 7 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
9 | 4, 7, 8 | konigsberglem1 28025 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘0) = 3 |
10 | 9 | breq2i 5066 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘0) ↔ 2 ∥ 3) |
11 | 6, 10 | mtbir 325 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0) |
12 | fveq2 6664 | . . . . . . 7 ⊢ (𝑥 = 0 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘0)) | |
13 | 12 | breq2d 5070 | . . . . . 6 ⊢ (𝑥 = 0 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
14 | 13 | notbid 320 | . . . . 5 ⊢ (𝑥 = 0 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
15 | 14 | elrab 3679 | . . . 4 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (0 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
16 | 5, 11, 15 | mpbir2an 709 | . . 3 ⊢ 0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
17 | 1nn0 11907 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
18 | 1le3 11843 | . . . . . 6 ⊢ 1 ≤ 3 | |
19 | elfz2nn0 12992 | . . . . . 6 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
20 | 17, 1, 18, 19 | mpbir3an 1337 | . . . . 5 ⊢ 1 ∈ (0...3) |
21 | 20, 4 | eleqtrri 2912 | . . . 4 ⊢ 1 ∈ 𝑉 |
22 | 4, 7, 8 | konigsberglem2 28026 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘1) = 3 |
23 | 22 | breq2i 5066 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘1) ↔ 2 ∥ 3) |
24 | 6, 23 | mtbir 325 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1) |
25 | fveq2 6664 | . . . . . . 7 ⊢ (𝑥 = 1 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘1)) | |
26 | 25 | breq2d 5070 | . . . . . 6 ⊢ (𝑥 = 1 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
27 | 26 | notbid 320 | . . . . 5 ⊢ (𝑥 = 1 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
28 | 27 | elrab 3679 | . . . 4 ⊢ (1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (1 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
29 | 21, 24, 28 | mpbir2an 709 | . . 3 ⊢ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
30 | 3re 11711 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
31 | 30 | leidi 11168 | . . . . . 6 ⊢ 3 ≤ 3 |
32 | elfz2nn0 12992 | . . . . . 6 ⊢ (3 ∈ (0...3) ↔ (3 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 ≤ 3)) | |
33 | 1, 1, 31, 32 | mpbir3an 1337 | . . . . 5 ⊢ 3 ∈ (0...3) |
34 | 33, 4 | eleqtrri 2912 | . . . 4 ⊢ 3 ∈ 𝑉 |
35 | 4, 7, 8 | konigsberglem3 28027 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘3) = 3 |
36 | 35 | breq2i 5066 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘3) ↔ 2 ∥ 3) |
37 | 6, 36 | mtbir 325 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3) |
38 | fveq2 6664 | . . . . . . 7 ⊢ (𝑥 = 3 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘3)) | |
39 | 38 | breq2d 5070 | . . . . . 6 ⊢ (𝑥 = 3 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
40 | 39 | notbid 320 | . . . . 5 ⊢ (𝑥 = 3 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
41 | 40 | elrab 3679 | . . . 4 ⊢ (3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (3 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
42 | 34, 37, 41 | mpbir2an 709 | . . 3 ⊢ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
43 | 16, 29, 42 | 3pm3.2i 1335 | . 2 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
44 | c0ex 10629 | . . 3 ⊢ 0 ∈ V | |
45 | 1ex 10631 | . . 3 ⊢ 1 ∈ V | |
46 | 3ex 11713 | . . 3 ⊢ 3 ∈ V | |
47 | 44, 45, 46 | tpss 4761 | . 2 ⊢ ((0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
48 | 43, 47 | mpbi 232 | 1 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 {cpr 4562 {ctp 4564 〈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 〈“cs7 14202 ∥ cdvds 15601 VtxDegcvtxdg 27241 |
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-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-xadd 12502 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 df-dvds 15602 df-vtx 26777 df-iedg 26778 df-vtxdg 27242 |
This theorem is referenced by: konigsberglem5 28029 |
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