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Mirrors > Home > MPE Home > Th. List > konigsberglem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for konigsberg 27963: 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 11903 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
2 | 0elfz 12992 | . . . . . 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 2909 | . . . 4 ⊢ 0 ∈ 𝑉 |
6 | n2dvds3 15709 | . . . . 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 27958 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘0) = 3 |
10 | 9 | breq2i 5065 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘0) ↔ 2 ∥ 3) |
11 | 6, 10 | mtbir 324 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0) |
12 | fveq2 6663 | . . . . . . 7 ⊢ (𝑥 = 0 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘0)) | |
13 | 12 | breq2d 5069 | . . . . . 6 ⊢ (𝑥 = 0 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
14 | 13 | notbid 319 | . . . . 5 ⊢ (𝑥 = 0 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
15 | 14 | elrab 3677 | . . . 4 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (0 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘0))) |
16 | 5, 11, 15 | mpbir2an 707 | . . 3 ⊢ 0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
17 | 1nn0 11901 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
18 | 1le3 11837 | . . . . . 6 ⊢ 1 ≤ 3 | |
19 | elfz2nn0 12986 | . . . . . 6 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3)) | |
20 | 17, 1, 18, 19 | mpbir3an 1333 | . . . . 5 ⊢ 1 ∈ (0...3) |
21 | 20, 4 | eleqtrri 2909 | . . . 4 ⊢ 1 ∈ 𝑉 |
22 | 4, 7, 8 | konigsberglem2 27959 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘1) = 3 |
23 | 22 | breq2i 5065 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘1) ↔ 2 ∥ 3) |
24 | 6, 23 | mtbir 324 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1) |
25 | fveq2 6663 | . . . . . . 7 ⊢ (𝑥 = 1 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘1)) | |
26 | 25 | breq2d 5069 | . . . . . 6 ⊢ (𝑥 = 1 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
27 | 26 | notbid 319 | . . . . 5 ⊢ (𝑥 = 1 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
28 | 27 | elrab 3677 | . . . 4 ⊢ (1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (1 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘1))) |
29 | 21, 24, 28 | mpbir2an 707 | . . 3 ⊢ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
30 | 3re 11705 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
31 | 30 | leidi 11162 | . . . . . 6 ⊢ 3 ≤ 3 |
32 | elfz2nn0 12986 | . . . . . 6 ⊢ (3 ∈ (0...3) ↔ (3 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 ≤ 3)) | |
33 | 1, 1, 31, 32 | mpbir3an 1333 | . . . . 5 ⊢ 3 ∈ (0...3) |
34 | 33, 4 | eleqtrri 2909 | . . . 4 ⊢ 3 ∈ 𝑉 |
35 | 4, 7, 8 | konigsberglem3 27960 | . . . . . 6 ⊢ ((VtxDeg‘𝐺)‘3) = 3 |
36 | 35 | breq2i 5065 | . . . . 5 ⊢ (2 ∥ ((VtxDeg‘𝐺)‘3) ↔ 2 ∥ 3) |
37 | 6, 36 | mtbir 324 | . . . 4 ⊢ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3) |
38 | fveq2 6663 | . . . . . . 7 ⊢ (𝑥 = 3 → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘3)) | |
39 | 38 | breq2d 5069 | . . . . . 6 ⊢ (𝑥 = 3 → (2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
40 | 39 | notbid 319 | . . . . 5 ⊢ (𝑥 = 3 → (¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥) ↔ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
41 | 40 | elrab 3677 | . . . 4 ⊢ (3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ↔ (3 ∈ 𝑉 ∧ ¬ 2 ∥ ((VtxDeg‘𝐺)‘3))) |
42 | 34, 37, 41 | mpbir2an 707 | . . 3 ⊢ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
43 | 16, 29, 42 | 3pm3.2i 1331 | . 2 ⊢ (0 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 1 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∧ 3 ∈ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
44 | c0ex 10623 | . . 3 ⊢ 0 ∈ V | |
45 | 1ex 10625 | . . 3 ⊢ 1 ∈ V | |
46 | 3ex 11707 | . . 3 ⊢ 3 ∈ V | |
47 | 44, 45, 46 | tpss 4760 | . 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 231 | 1 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 {cpr 4559 {ctp 4561 〈cop 4563 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 ≤ cle 10664 2c2 11680 3c3 11681 ℕ0cn0 11885 ...cfz 12880 〈“cs7 14196 ∥ cdvds 15595 VtxDegcvtxdg 27174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-s4 14200 df-s5 14201 df-s6 14202 df-s7 14203 df-dvds 15596 df-vtx 26710 df-iedg 26711 df-vtxdg 27175 |
This theorem is referenced by: konigsberglem5 27962 |
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