![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 |
---|---|
friendship.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
friendship | ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1191 | . . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐺 ∈ FriendGraph ) | |
2 | simpr3 1193 | . . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝑉 ∈ Fin) | |
3 | simpl 481 | . . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 3 < (♯‘𝑉)) | |
4 | friendship.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | friendshipgt3 30250 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) |
6 | 1, 2, 3, 5 | syl3anc 1368 | . . 3 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) |
7 | 6 | ex 411 | . 2 ⊢ (3 < (♯‘𝑉) → ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
8 | hashcl 14345 | . . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0) | |
9 | simplr 767 | . . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 𝑉 ∈ Fin) | |
10 | hashge1 14378 | . . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 1 ≤ (♯‘𝑉)) | |
11 | 10 | ad2ant2l 744 | . . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 1 ≤ (♯‘𝑉)) |
12 | nn0re 12509 | . . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ) | |
13 | 3re 12320 | . . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ | |
14 | lenlt 11320 | . . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑉) ∈ ℝ ∧ 3 ∈ ℝ) → ((♯‘𝑉) ≤ 3 ↔ ¬ 3 < (♯‘𝑉))) | |
15 | 12, 13, 14 | sylancl 584 | . . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑉) ∈ ℕ0 → ((♯‘𝑉) ≤ 3 ↔ ¬ 3 < (♯‘𝑉))) |
16 | 15 | biimprd 247 | . . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℕ0 → (¬ 3 < (♯‘𝑉) → (♯‘𝑉) ≤ 3)) |
17 | 16 | adantr 479 | . . . . . . . . . . . . . 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 479 | . . . . . . . . . . . 12 ⊢ ((¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅) → (((♯‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) → (♯‘𝑉) ≤ 3)) |
20 | 19 | impcom 406 | . . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → (♯‘𝑉) ≤ 3) |
21 | 9, 11, 20 | 3jca 1125 | . . . . . . . . . 10 ⊢ ((((♯‘𝑉) ∈ ℕ0 ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → (𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3)) |
22 | 21 | exp31 418 | . . . . . . . . 9 ⊢ ((♯‘𝑉) ∈ ℕ0 → (𝑉 ∈ Fin → ((¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3)))) |
23 | 8, 22 | mpcom 38 | . . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3))) |
24 | 23 | impcom 406 | . . . . . . 7 ⊢ (((¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅) ∧ 𝑉 ∈ Fin) → (𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3)) |
25 | hash1to3 14482 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐})) | |
26 | vex 3467 | . . . . . . . . . 10 ⊢ 𝑎 ∈ V | |
27 | eqid 2725 | . . . . . . . . . . 11 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
28 | 4, 27 | 1to3vfriendship 30133 | . . . . . . . . . 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 688 | . . . . . . . . 9 ⊢ ((𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) |
30 | 29 | exlimiv 1925 | . . . . . . . 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 1927 | . . . . . . 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 418 | . . . . 5 ⊢ (¬ 3 < (♯‘𝑉) → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))) |
34 | 33 | com14 96 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ 3 < (♯‘𝑉) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))) |
35 | 34 | 3imp 1108 | . . 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 182 | 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 205 ∧ wa 394 ∨ w3o 1083 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ∖ cdif 3937 ∅c0 4318 {csn 4624 {cpr 4626 {ctp 4628 class class class wbr 5143 ‘cfv 6542 Fincfn 8960 ℝcr 11135 1c1 11137 < clt 11276 ≤ cle 11277 3c3 12296 ℕ0cn0 12500 ♯chash 14319 Vtxcvtx 28851 Edgcedg 28902 FriendGraph cfrgr 30110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-ac2 10484 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-3o 8485 df-oadd 8487 df-er 8721 df-ec 8723 df-qs 8727 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-oi 9531 df-dju 9922 df-card 9960 df-ac 10137 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-rp 13005 df-xadd 13123 df-ico 13360 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-seq 13997 df-exp 14057 df-hash 14320 df-word 14495 df-lsw 14543 df-concat 14551 df-s1 14576 df-substr 14621 df-pfx 14651 df-reps 14749 df-csh 14769 df-s2 14829 df-s3 14830 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-dvds 16229 df-gcd 16467 df-prm 16640 df-phi 16732 df-vtx 28853 df-iedg 28854 df-edg 28903 df-uhgr 28913 df-ushgr 28914 df-upgr 28937 df-umgr 28938 df-uspgr 29005 df-usgr 29006 df-fusgr 29172 df-nbgr 29188 df-vtxdg 29322 df-rgr 29413 df-rusgr 29414 df-wlks 29455 df-wlkson 29456 df-trls 29548 df-trlson 29549 df-pths 29572 df-spths 29573 df-pthson 29574 df-spthson 29575 df-wwlks 29683 df-wwlksn 29684 df-wwlksnon 29685 df-wspthsn 29686 df-wspthsnon 29687 df-clwwlk 29834 df-clwwlkn 29877 df-clwwlknon 29940 df-conngr 30039 df-frgr 30111 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |