Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > av-friendship | Structured version Visualization version GIF version |
Description: The friendship theorem: In every finite (nonempty) friendship graph there is a vertex which is adjacent to all other vertices. This is Metamath 100 proof #83. (Contributed by Alexander van der Vekens, 9-Oct-2018.) |
Ref | Expression |
---|---|
av-friendship.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
av-friendship | ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1060 | . . . 4 ⊢ ((3 < (#‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐺 ∈ FriendGraph ) | |
2 | simpr3 1062 | . . . 4 ⊢ ((3 < (#‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝑉 ∈ Fin) | |
3 | simpl 472 | . . . 4 ⊢ ((3 < (#‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 3 < (#‘𝑉)) | |
4 | av-friendship.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | av-friendshipgt3 41552 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) |
6 | 1, 2, 3, 5 | syl3anc 1318 | . . 3 ⊢ ((3 < (#‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) |
7 | 6 | ex 449 | . 2 ⊢ (3 < (#‘𝑉) → ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
8 | hashcl 13009 | . . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0) | |
9 | simplr 788 | . . . . . . . . . . 11 ⊢ ((((#‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅)) → 𝑉 ∈ Fin) | |
10 | hashge1 13039 | . . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 1 ≤ (#‘𝑉)) | |
11 | 10 | ad2ant2l 778 | . . . . . . . . . . 11 ⊢ ((((#‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅)) → 1 ≤ (#‘𝑉)) |
12 | nn0re 11178 | . . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℝ) | |
13 | 3re 10971 | . . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ | |
14 | lenlt 9995 | . . . . . . . . . . . . . . . . 17 ⊢ (((#‘𝑉) ∈ ℝ ∧ 3 ∈ ℝ) → ((#‘𝑉) ≤ 3 ↔ ¬ 3 < (#‘𝑉))) | |
15 | 12, 13, 14 | sylancl 693 | . . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑉) ∈ ℕ0 → ((#‘𝑉) ≤ 3 ↔ ¬ 3 < (#‘𝑉))) |
16 | 15 | biimprd 237 | . . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℕ0 → (¬ 3 < (#‘𝑉) → (#‘𝑉) ≤ 3)) |
17 | 16 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ (((#‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) → (¬ 3 < (#‘𝑉) → (#‘𝑉) ≤ 3)) |
18 | 17 | com12 32 | . . . . . . . . . . . . 13 ⊢ (¬ 3 < (#‘𝑉) → (((#‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) → (#‘𝑉) ≤ 3)) |
19 | 18 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅) → (((#‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) → (#‘𝑉) ≤ 3)) |
20 | 19 | impcom 445 | . . . . . . . . . . 11 ⊢ ((((#‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅)) → (#‘𝑉) ≤ 3) |
21 | 9, 11, 20 | 3jca 1235 | . . . . . . . . . 10 ⊢ ((((#‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅)) → (𝑉 ∈ Fin ∧ 1 ≤ (#‘𝑉) ∧ (#‘𝑉) ≤ 3)) |
22 | 21 | exp31 628 | . . . . . . . . 9 ⊢ ((#‘𝑉) ∈ ℕ0 → (𝑉 ∈ Fin → ((¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin ∧ 1 ≤ (#‘𝑉) ∧ (#‘𝑉) ≤ 3)))) |
23 | 8, 22 | mpcom 37 | . . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin ∧ 1 ≤ (#‘𝑉) ∧ (#‘𝑉) ≤ 3))) |
24 | 23 | impcom 445 | . . . . . . 7 ⊢ (((¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅) ∧ 𝑉 ∈ Fin) → (𝑉 ∈ Fin ∧ 1 ≤ (#‘𝑉) ∧ (#‘𝑉) ≤ 3)) |
25 | hash1to3 13128 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (#‘𝑉) ∧ (#‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) | |
26 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑎 ∈ V | |
27 | eqid 2610 | . . . . . . . . . . 11 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
28 | 4, 27 | 1to3vfriendship-av 41451 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ V ∧ (𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
29 | 26, 28 | mpan 702 | . . . . . . . . 9 ⊢ ((𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
30 | 29 | exlimiv 1845 | . . . . . . . 8 ⊢ (∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
31 | 30 | exlimivv 1847 | . . . . . . 7 ⊢ (∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
32 | 24, 25, 31 | 3syl 18 | . . . . . 6 ⊢ (((¬ 3 < (#‘𝑉) ∧ 𝑉 ≠ ∅) ∧ 𝑉 ∈ Fin) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
33 | 32 | exp31 628 | . . . . 5 ⊢ (¬ 3 < (#‘𝑉) → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))) |
34 | 33 | com14 94 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ 3 < (#‘𝑉) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))) |
35 | 34 | 3imp 1249 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → (¬ 3 < (#‘𝑉) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
36 | 35 | com12 32 | . 2 ⊢ (¬ 3 < (#‘𝑉) → ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
37 | 7, 36 | pm2.61i 175 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 {csn 4125 {cpr 4127 {ctp 4129 class class class wbr 4583 ‘cfv 5804 Fincfn 7841 ℝcr 9814 1c1 9816 < clt 9953 ≤ cle 9954 3c3 10948 ℕ0cn0 11169 #chash 12979 Vtxcvtx 25673 Edgcedga 25792 FriendGraph cfrgr 41428 |
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-inf2 8421 ax-ac2 9168 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 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-disj 4554 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-se 4998 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-isom 5813 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-2o 7448 df-3o 7449 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-ac 8822 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-rp 11709 df-xadd 11823 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-reps 13161 df-csh 13386 df-s2 13444 df-s3 13445 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-phi 15309 df-vtx 25675 df-iedg 25676 df-uhgr 25724 df-ushgr 25725 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-fusgr 40536 df-nbgr 40554 df-vtxdg 40682 df-rgr 40757 df-rusgr 40758 df-1wlks 40800 df-wlks 40801 df-wlkson 40802 df-trls 40901 df-trlson 40902 df-pths 40923 df-spths 40924 df-pthson 40925 df-spthson 40926 df-wwlks 41033 df-wwlksn 41034 df-wwlksnon 41035 df-wspthsn 41036 df-wspthsnon 41037 df-clwwlks 41185 df-clwwlksn 41186 df-conngr 41354 df-frgr 41429 |
This theorem is referenced by: (None) |
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