friendship - Metamath Proof Explorer

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Theorem friendship 30385

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 1195	. . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐺 ∈ FriendGraph )
2		simpr3 1197	. . . 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 30384	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))
6	1, 2, 3, 5	syl3anc 1373	. . 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 14379	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ₀)
9		simplr 768	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 𝑉 ∈ Fin)
10		hashge1 14412	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 1 ≤ (♯‘𝑉))
11	10	ad2ant2l 746	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 1 ≤ (♯‘𝑉))
12		nn0re 12515	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (♯‘𝑉) ∈ ℝ)
13		3re 12325	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
14		lenlt 11318	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑉) ∈ ℝ ∧ 3 ∈ ℝ) → ((♯‘𝑉) ≤ 3 ↔ ¬ 3 < (♯‘𝑉)))
15	12, 13, 14	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑉) ∈ ℕ₀ → ((♯‘𝑉) ≤ 3 ↔ ¬ 3 < (♯‘𝑉)))
16	15	biimprd 248	. . . . . . . . . . . . . . 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 1128	. . . . . . . . . 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 14515	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))
26		vex 3468	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
27		eqid 2736	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
28	4, 27	1to3vfriendship 30267	. . . . . . . . . 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 690	. . . . . . . . 9 ⊢ ((𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
30	29	exlimiv 1930	. . . . . . . 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 1932	. . . . . . 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 1110	. . 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 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 Vcvv 3464 ∖ cdif 3928 ∅c0 4313 {csn 4606 {cpr 4608 {ctp 4610 class class class wbr 5124 ‘cfv 6536 Fincfn 8964 ℝcr 11133 1c1 11135 < clt 11274 ≤ cle 11275 3c3 12301 ℕ₀cn0 12506 ♯chash 14353 Vtxcvtx 28980 Edgcedg 29031 FriendGraph cfrgr 30244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-ac2 10482 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-3o 8487 df-oadd 8489 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-ac 10135 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-rp 13014 df-xadd 13134 df-ico 13373 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-hash 14354 df-word 14537 df-lsw 14586 df-concat 14594 df-s1 14619 df-substr 14664 df-pfx 14694 df-reps 14792 df-csh 14812 df-s2 14872 df-s3 14873 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-sum 15708 df-dvds 16278 df-gcd 16519 df-prm 16696 df-phi 16790 df-vtx 28982 df-iedg 28983 df-edg 29032 df-uhgr 29042 df-ushgr 29043 df-upgr 29066 df-umgr 29067 df-uspgr 29134 df-usgr 29135 df-fusgr 29301 df-nbgr 29317 df-vtxdg 29451 df-rgr 29542 df-rusgr 29543 df-wlks 29584 df-wlkson 29585 df-trls 29677 df-trlson 29678 df-pths 29701 df-spths 29702 df-pthson 29703 df-spthson 29704 df-wwlks 29817 df-wwlksn 29818 df-wwlksnon 29819 df-wspthsn 29820 df-wspthsnon 29821 df-clwwlk 29968 df-clwwlkn 30011 df-clwwlknon 30074 df-conngr 30173 df-frgr 30245

This theorem is referenced by: (None)

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