![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > konigsberglem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for konigsberg 27596: 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 11596 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 12687 | . . . . . 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 2875 | . . . 4 ⊢ 0 ∈ 𝑉 |
6 | n2dvds3 15440 | . . . . 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 27591 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘0) = 3 |
10 | 9 | breq2i 4849 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘0) ↔ 2 ∥ 3) |
11 | 6, 10 | mtbir 315 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0) |
12 | fveq2 6409 | . . . . . . 7 ⊢ (𝑥 = 0 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘0)) | |
13 | 12 | breq2d 4853 | . . . . . 6 ⊢ (𝑥 = 0 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
14 | 13 | notbid 310 | . . . . 5 ⊢ (𝑥 = 0 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
15 | 14 | elrab 3554 | . . . 4 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (0 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
16 | 5, 11, 15 | mpbir2an 703 | . . 3 ⊢ 0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
17 | 1nn0 11594 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
18 | 1le3 11528 | . . . . . 6 ⊢ 1 ≤ 3 | |
19 | elfz2nn0 12681 | . . . . . 6 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
20 | 17, 1, 18, 19 | mpbir3an 1442 | . . . . 5 ⊢ 1 ∈ (0...3) |
21 | 20, 4 | eleqtrri 2875 | . . . 4 ⊢ 1 ∈ 𝑉 |
22 | 4, 7, 8 | konigsberglem2 27592 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘1) = 3 |
23 | 22 | breq2i 4849 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘1) ↔ 2 ∥ 3) |
24 | 6, 23 | mtbir 315 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1) |
25 | fveq2 6409 | . . . . . . 7 ⊢ (𝑥 = 1 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘1)) | |
26 | 25 | breq2d 4853 | . . . . . 6 ⊢ (𝑥 = 1 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
27 | 26 | notbid 310 | . . . . 5 ⊢ (𝑥 = 1 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
28 | 27 | elrab 3554 | . . . 4 ⊢ (1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (1 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
29 | 21, 24, 28 | mpbir2an 703 | . . 3 ⊢ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
30 | 3re 11389 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
31 | 30 | leidi 10852 | . . . . . 6 ⊢ 3 ≤ 3 |
32 | elfz2nn0 12681 | . . . . . 6 ⊢ (3 ∈ (0...3) ↔ (3 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 ≤ 3)) | |
33 | 1, 1, 31, 32 | mpbir3an 1442 | . . . . 5 ⊢ 3 ∈ (0...3) |
34 | 33, 4 | eleqtrri 2875 | . . . 4 ⊢ 3 ∈ 𝑉 |
35 | 4, 7, 8 | konigsberglem3 27593 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘3) = 3 |
36 | 35 | breq2i 4849 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘3) ↔ 2 ∥ 3) |
37 | 6, 36 | mtbir 315 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3) |
38 | fveq2 6409 | . . . . . . 7 ⊢ (𝑥 = 3 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘3)) | |
39 | 38 | breq2d 4853 | . . . . . 6 ⊢ (𝑥 = 3 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
40 | 39 | notbid 310 | . . . . 5 ⊢ (𝑥 = 3 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
41 | 40 | elrab 3554 | . . . 4 ⊢ (3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (3 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
42 | 34, 37, 41 | mpbir2an 703 | . . 3 ⊢ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
43 | 16, 29, 42 | 3pm3.2i 1439 | . 2 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
44 | c0ex 10320 | . . 3 ⊢ 0 ∈ V | |
45 | 1ex 10322 | . . 3 ⊢ 1 ∈ V | |
46 | 3ex 11392 | . . 3 ⊢ 3 ∈ V | |
47 | 44, 45, 46 | tpss 4552 | . 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 222 | 1 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 {crab 3091 ⊆ wss 3767 {cpr 4368 {ctp 4370 〈cop 4372 class class class wbr 4841 ‘cfv 6099 (class class class)co 6876 0cc0 10222 1c1 10223 ≤ cle 10362 2c2 11364 3c3 11365 ℕ0cn0 11576 ...cfz 12576 〈“cs7 13928 ∥ cdvds 15316 VtxDegcvtxdg 26707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-card 9049 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-n0 11577 df-xnn0 11649 df-z 11663 df-uz 11927 df-xadd 12190 df-fz 12577 df-fzo 12717 df-hash 13367 df-word 13531 df-concat 13587 df-s1 13612 df-s2 13930 df-s3 13931 df-s4 13932 df-s5 13933 df-s6 13934 df-s7 13935 df-dvds 15317 df-vtx 26225 df-iedg 26226 df-vtxdg 26708 |
This theorem is referenced by: konigsberglem5 27595 |
Copyright terms: Public domain | W3C validator |