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Mirrors > Home > MPE Home > Th. List > konigsberglem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for konigsberg 27119: The set of vertices of odd degree is greater than 2. (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 |
---|---|
konigsberglem5 | ⊢ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | konigsberg.v | . . 3 ⊢ 𝑉 = (0...3) | |
2 | konigsberg.e | . . 3 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
3 | konigsberg.g | . . 3 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
4 | 1, 2, 3 | konigsberglem4 27117 | . 2 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
5 | 1 | ovexi 6679 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 5 | rabex 4813 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ V |
7 | hashss 13197 | . . 3 ⊢ (({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ V ∧ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) → (#‘{0, 1, 3}) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) | |
8 | 6, 7 | mpan 706 | . 2 ⊢ ({0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} → (#‘{0, 1, 3}) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
9 | 0ne1 11088 | . . . . . 6 ⊢ 0 ≠ 1 | |
10 | 1re 10039 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
11 | 1lt3 11196 | . . . . . . 7 ⊢ 1 < 3 | |
12 | 10, 11 | ltneii 10150 | . . . . . 6 ⊢ 1 ≠ 3 |
13 | 3ne0 11115 | . . . . . 6 ⊢ 3 ≠ 0 | |
14 | 9, 12, 13 | 3pm3.2i 1239 | . . . . 5 ⊢ (0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) |
15 | c0ex 10034 | . . . . . 6 ⊢ 0 ∈ V | |
16 | 1ex 10035 | . . . . . 6 ⊢ 1 ∈ V | |
17 | 3ex 11096 | . . . . . 6 ⊢ 3 ∈ V | |
18 | hashtpg 13267 | . . . . . 6 ⊢ ((0 ∈ V ∧ 1 ∈ V ∧ 3 ∈ V) → ((0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) ↔ (#‘{0, 1, 3}) = 3)) | |
19 | 15, 16, 17, 18 | mp3an 1424 | . . . . 5 ⊢ ((0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) ↔ (#‘{0, 1, 3}) = 3) |
20 | 14, 19 | mpbi 220 | . . . 4 ⊢ (#‘{0, 1, 3}) = 3 |
21 | 20 | breq1i 4660 | . . 3 ⊢ ((#‘{0, 1, 3}) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 3 ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
22 | df-3 11080 | . . . . 5 ⊢ 3 = (2 + 1) | |
23 | 22 | breq1i 4660 | . . . 4 ⊢ (3 ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ (2 + 1) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
24 | 2z 11409 | . . . . 5 ⊢ 2 ∈ ℤ | |
25 | fzfi 12771 | . . . . . . . 8 ⊢ (0...3) ∈ Fin | |
26 | 1, 25 | eqeltri 2697 | . . . . . . 7 ⊢ 𝑉 ∈ Fin |
27 | rabfi 8185 | . . . . . . 7 ⊢ (𝑉 ∈ Fin → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ Fin) | |
28 | hashcl 13147 | . . . . . . 7 ⊢ ({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ Fin → (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ ℕ0) | |
29 | 26, 27, 28 | mp2b 10 | . . . . . 6 ⊢ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ ℕ0 |
30 | 29 | nn0zi 11402 | . . . . 5 ⊢ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ ℤ |
31 | zltp1le 11427 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ ℤ) → (2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ (2 + 1) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))) | |
32 | 24, 30, 31 | mp2an 708 | . . . 4 ⊢ (2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ (2 + 1) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
33 | 23, 32 | sylbb2 228 | . . 3 ⊢ (3 ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) → 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
34 | 21, 33 | sylbi 207 | . 2 ⊢ ((#‘{0, 1, 3}) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) → 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
35 | 4, 8, 34 | mp2b 10 | 1 ⊢ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ⊆ wss 3574 {cpr 4179 {ctp 4181 〈cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 2c2 11070 3c3 11071 ℕ0cn0 11292 ℤcz 11377 ...cfz 12326 #chash 13117 〈“cs7 13591 ∥ cdvds 14983 VtxDegcvtxdg 26361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-xadd 11947 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-s4 13595 df-s5 13596 df-s6 13597 df-s7 13598 df-dvds 14984 df-vtx 25876 df-iedg 25877 df-vtxdg 26362 |
This theorem is referenced by: konigsberg 27119 |
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