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Mirrors > Home > MPE Home > Th. List > konigsberglem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for konigsberg 28036: 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 28034 | . 2 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
5 | 1 | ovexi 7190 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 5 | rabex 5235 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ V |
7 | hashss 13771 | . . 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 688 | . 2 ⊢ ({0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} → (♯‘{0, 1, 3}) ≤ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
9 | 0ne1 11709 | . . . . . 6 ⊢ 0 ≠ 1 | |
10 | 1re 10641 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
11 | 1lt3 11811 | . . . . . . 7 ⊢ 1 < 3 | |
12 | 10, 11 | ltneii 10753 | . . . . . 6 ⊢ 1 ≠ 3 |
13 | 3ne0 11744 | . . . . . 6 ⊢ 3 ≠ 0 | |
14 | 9, 12, 13 | 3pm3.2i 1335 | . . . . 5 ⊢ (0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) |
15 | c0ex 10635 | . . . . . 6 ⊢ 0 ∈ V | |
16 | 1ex 10637 | . . . . . 6 ⊢ 1 ∈ V | |
17 | 3ex 11720 | . . . . . 6 ⊢ 3 ∈ V | |
18 | hashtpg 13844 | . . . . . 6 ⊢ ((0 ∈ V ∧ 1 ∈ V ∧ 3 ∈ V) → ((0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) ↔ (♯‘{0, 1, 3}) = 3)) | |
19 | 15, 16, 17, 18 | mp3an 1457 | . . . . 5 ⊢ ((0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) ↔ (♯‘{0, 1, 3}) = 3) |
20 | 14, 19 | mpbi 232 | . . . 4 ⊢ (♯‘{0, 1, 3}) = 3 |
21 | 20 | breq1i 5073 | . . 3 ⊢ ((♯‘{0, 1, 3}) ≤ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 3 ≤ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
22 | df-3 11702 | . . . . 5 ⊢ 3 = (2 + 1) | |
23 | 22 | breq1i 5073 | . . . 4 ⊢ (3 ≤ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ (2 + 1) ≤ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
24 | 2z 12015 | . . . . 5 ⊢ 2 ∈ ℤ | |
25 | fzfi 13341 | . . . . . . . 8 ⊢ (0...3) ∈ Fin | |
26 | 1, 25 | eqeltri 2909 | . . . . . . 7 ⊢ 𝑉 ∈ Fin |
27 | rabfi 8743 | . . . . . . 7 ⊢ (𝑉 ∈ Fin → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ Fin) | |
28 | hashcl 13718 | . . . . . . 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 12008 | . . . . 5 ⊢ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ ℤ |
31 | zltp1le 12033 | . . . . 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 690 | . . . 4 ⊢ (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ (2 + 1) ≤ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
33 | 23, 32 | sylbb2 240 | . . 3 ⊢ (3 ≤ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) → 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
34 | 21, 33 | sylbi 219 | . 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 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {crab 3142 Vcvv 3494 ⊆ wss 3936 {cpr 4569 {ctp 4571 〈cop 4573 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 2c2 11693 3c3 11694 ℕ0cn0 11898 ℤcz 11982 ...cfz 12893 ♯chash 13691 〈“cs7 14208 ∥ cdvds 15607 VtxDegcvtxdg 27247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-xadd 12509 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-s4 14212 df-s5 14213 df-s6 14214 df-s7 14215 df-dvds 15608 df-vtx 26783 df-iedg 26784 df-vtxdg 27248 |
This theorem is referenced by: konigsberg 28036 |
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