friendship - Metamath Proof Explorer

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Theorem friendship 28664

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.)

**Hypothesis**
Ref	Expression
friendship.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
friendship	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))

Distinct variable groups: 𝑣,𝐺,𝑤 𝑣,𝑉,𝑤

Proof of Theorem friendship Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr1 1192	. . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐺 ∈ FriendGraph )
2		simpr3 1194	. . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝑉 ∈ Fin)
3		simpl 482	. . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 3 < (♯‘𝑉))
4		friendship.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	friendshipgt3 28663	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))
6	1, 2, 3, 5	syl3anc 1369	. . 3 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))
7	6	ex 412	. 2 ⊢ (3 < (♯‘𝑉) → ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
8		hashcl 13999	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ₀)
9		simplr 765	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 𝑉 ∈ Fin)
10		hashge1 14032	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 1 ≤ (♯‘𝑉))
11	10	ad2ant2l 742	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 1 ≤ (♯‘𝑉))
12		nn0re 12172	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (♯‘𝑉) ∈ ℝ)
13		3re 11983	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
14		lenlt 10984	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑉) ∈ ℝ ∧ 3 ∈ ℝ) → ((♯‘𝑉) ≤ 3 ↔ ¬ 3 < (♯‘𝑉)))
15	12, 13, 14	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑉) ∈ ℕ₀ → ((♯‘𝑉) ≤ 3 ↔ ¬ 3 < (♯‘𝑉)))
16	15	biimprd 247	. . . . . . . . . . . . . . 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 407	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → (♯‘𝑉) ≤ 3)
21	9, 11, 20	3jca 1126	. . . . . . . . . 10 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → (𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3))
22	21	exp31 419	. . . . . . . . 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 407	. . . . . . 7 ⊢ (((¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅) ∧ 𝑉 ∈ Fin) → (𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3))
25		hash1to3 14133	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))
26		vex 3426	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
27		eqid 2738	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
28	4, 27	1to3vfriendship 28546	. . . . . . . . . 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 686	. . . . . . . . 9 ⊢ ((𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
30	29	exlimiv 1934	. . . . . . . 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 1936	. . . . . . 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 419	. . . . 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 1109	. . 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 395 ∨ w3o 1084 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∖ cdif 3880 ∅c0 4253 {csn 4558 {cpr 4560 {ctp 4562 class class class wbr 5070 ‘cfv 6418 Fincfn 8691 ℝcr 10801 1c1 10803 < clt 10940 ≤ cle 10941 3c3 11959 ℕ₀cn0 12163 ♯chash 13972 Vtxcvtx 27269 Edgcedg 27320 FriendGraph cfrgr 28523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-ac2 10150 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-3o 8269 df-oadd 8271 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-ac 9803 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-xadd 12778 df-ico 13014 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-reps 14410 df-csh 14430 df-s2 14489 df-s3 14490 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-dvds 15892 df-gcd 16130 df-prm 16305 df-phi 16395 df-vtx 27271 df-iedg 27272 df-edg 27321 df-uhgr 27331 df-ushgr 27332 df-upgr 27355 df-umgr 27356 df-uspgr 27423 df-usgr 27424 df-fusgr 27587 df-nbgr 27603 df-vtxdg 27736 df-rgr 27827 df-rusgr 27828 df-wlks 27869 df-wlkson 27870 df-trls 27962 df-trlson 27963 df-pths 27985 df-spths 27986 df-pthson 27987 df-spthson 27988 df-wwlks 28096 df-wwlksn 28097 df-wwlksnon 28098 df-wspthsn 28099 df-wspthsnon 28100 df-clwwlk 28247 df-clwwlkn 28290 df-clwwlknon 28353 df-conngr 28452 df-frgr 28524

This theorem is referenced by: (None)

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