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Mirrors > Home > MPE Home > Th. List > Mathboxes > konigsberglem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for konigsberg-av 41427: 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-av.v | ⊢ 𝑉 = (0...3) |
konigsberg-av.e | ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 |
konigsberg-av.g | ⊢ 𝐺 = 〈𝑉, 𝐸〉 |
Ref | Expression |
---|---|
konigsberglem5 | ⊢ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | konigsberg-av.v | . . 3 ⊢ 𝑉 = (0...3) | |
2 | konigsberg-av.e | . . 3 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
3 | konigsberg-av.g | . . 3 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
4 | 1, 2, 3 | konigsberglem4 41425 | . 2 ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} |
5 | 1 | ovexi 6578 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 5 | rabex 4740 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ V |
7 | hashss 13058 | . . 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 702 | . 2 ⊢ ({0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} → (#‘{0, 1, 3}) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
9 | 0ne1 10965 | . . . . . 6 ⊢ 0 ≠ 1 | |
10 | 1re 9918 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
11 | 1lt3 11073 | . . . . . . 7 ⊢ 1 < 3 | |
12 | 10, 11 | ltneii 10029 | . . . . . 6 ⊢ 1 ≠ 3 |
13 | 3ne0 10992 | . . . . . 6 ⊢ 3 ≠ 0 | |
14 | 9, 12, 13 | 3pm3.2i 1232 | . . . . 5 ⊢ (0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) |
15 | c0ex 9913 | . . . . . 6 ⊢ 0 ∈ V | |
16 | 1ex 9914 | . . . . . 6 ⊢ 1 ∈ V | |
17 | 3ex 10973 | . . . . . 6 ⊢ 3 ∈ V | |
18 | hashtpg 13121 | . . . . . 6 ⊢ ((0 ∈ V ∧ 1 ∈ V ∧ 3 ∈ V) → ((0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) ↔ (#‘{0, 1, 3}) = 3)) | |
19 | 15, 16, 17, 18 | mp3an 1416 | . . . . 5 ⊢ ((0 ≠ 1 ∧ 1 ≠ 3 ∧ 3 ≠ 0) ↔ (#‘{0, 1, 3}) = 3) |
20 | 14, 19 | mpbi 219 | . . . 4 ⊢ (#‘{0, 1, 3}) = 3 |
21 | 20 | breq1i 4590 | . . 3 ⊢ ((#‘{0, 1, 3}) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 3 ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
22 | df-3 10957 | . . . . 5 ⊢ 3 = (2 + 1) | |
23 | 22 | breq1i 4590 | . . . 4 ⊢ (3 ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ (2 + 1) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
24 | 2z 11286 | . . . . 5 ⊢ 2 ∈ ℤ | |
25 | fzfi 12633 | . . . . . . . 8 ⊢ (0...3) ∈ Fin | |
26 | 1, 25 | eqeltri 2684 | . . . . . . 7 ⊢ 𝑉 ∈ Fin |
27 | rabfi 8070 | . . . . . . 7 ⊢ (𝑉 ∈ Fin → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} ∈ Fin) | |
28 | hashcl 13009 | . . . . . . 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 11279 | . . . . 5 ⊢ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ ℤ |
31 | zltp1le 11304 | . . . . 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 704 | . . . 4 ⊢ (2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ (2 + 1) ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
33 | 23, 32 | sylbb2 227 | . . 3 ⊢ (3 ≤ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) → 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
34 | 21, 33 | sylbi 206 | . 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 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ⊆ wss 3540 {cpr 4127 {ctp 4129 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 2c2 10947 3c3 10948 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 #chash 12979 〈“cs7 13442 ∥ cdvds 14821 VtxDegcvtxdg 40681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-s4 13446 df-s5 13447 df-s6 13448 df-s7 13449 df-dvds 14822 df-vtx 25675 df-iedg 25676 df-vtxdg 40682 |
This theorem is referenced by: konigsberg-av 41427 |
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