![]() |
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 30139: 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 12523 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 13633 | . . . . . 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 2824 | . . . 4 ⊢ 0 ∈ 𝑉 |
6 | n2dvds3 16351 | . . . . 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 30134 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘0) = 3 |
10 | 9 | breq2i 5157 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘0) ↔ 2 ∥ 3) |
11 | 6, 10 | mtbir 322 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0) |
12 | fveq2 6896 | . . . . . . 7 ⊢ (𝑥 = 0 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘0)) | |
13 | 12 | breq2d 5161 | . . . . . 6 ⊢ (𝑥 = 0 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
14 | 13 | notbid 317 | . . . . 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 12521 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
18 | 1le3 12457 | . . . . . 6 ⊢ 1 ≤ 3 | |
19 | elfz2nn0 13627 | . . . . . 6 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
20 | 17, 1, 18, 19 | mpbir3an 1338 | . . . . 5 ⊢ 1 ∈ (0...3) |
21 | 20, 4 | eleqtrri 2824 | . . . 4 ⊢ 1 ∈ 𝑉 |
22 | 4, 7, 8 | konigsberglem2 30135 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘1) = 3 |
23 | 22 | breq2i 5157 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘1) ↔ 2 ∥ 3) |
24 | 6, 23 | mtbir 322 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1) |
25 | fveq2 6896 | . . . . . . 7 ⊢ (𝑥 = 1 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘1)) | |
26 | 25 | breq2d 5161 | . . . . . 6 ⊢ (𝑥 = 1 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
27 | 26 | notbid 317 | . . . . 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 12325 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
31 | 30 | leidi 11780 | . . . . . 6 ⊢ 3 ≤ 3 |
32 | elfz2nn0 13627 | . . . . . 6 ⊢ (3 ∈ (0...3) ↔ (3 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 ≤ 3)) | |
33 | 1, 1, 31, 32 | mpbir3an 1338 | . . . . 5 ⊢ 3 ∈ (0...3) |
34 | 33, 4 | eleqtrri 2824 | . . . 4 ⊢ 3 ∈ 𝑉 |
35 | 4, 7, 8 | konigsberglem3 30136 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘3) = 3 |
36 | 35 | breq2i 5157 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘3) ↔ 2 ∥ 3) |
37 | 6, 36 | mtbir 322 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3) |
38 | fveq2 6896 | . . . . . . 7 ⊢ (𝑥 = 3 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘3)) | |
39 | 38 | breq2d 5161 | . . . . . 6 ⊢ (𝑥 = 3 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
40 | 39 | notbid 317 | . . . . 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 1336 | . 2 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
44 | c0ex 11240 | . . 3 ⊢ 0 ∈ V | |
45 | 1ex 11242 | . . 3 ⊢ 1 ∈ V | |
46 | 3ex 12327 | . . 3 ⊢ 3 ∈ V | |
47 | 44, 45, 46 | tpss 4840 | . 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 229 | 1 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3418 ⊆ wss 3944 {cpr 4632 {ctp 4634 〈cop 4636 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 ≤ cle 11281 2c2 12300 3c3 12301 ℕ0cn0 12505 ...cfz 13519 〈“cs7 14833 ∥ cdvds 16234 VtxDegcvtxdg 29351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-xadd 13128 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-concat 14557 df-s1 14582 df-s2 14835 df-s3 14836 df-s4 14837 df-s5 14838 df-s6 14839 df-s7 14840 df-dvds 16235 df-vtx 28883 df-iedg 28884 df-vtxdg 29352 |
This theorem is referenced by: konigsberglem5 30138 |
Copyright terms: Public domain | W3C validator |