friendship - Metamath Proof Explorer

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Theorem friendship 30161

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 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 482	. . . 4 ⊢ ((3 < (♯‘𝑉) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 3 < (♯‘𝑉))
4		friendship.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	friendshipgt3 30160	. . . 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 412	. 2 ⊢ (3 < (♯‘𝑉) → ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
8		hashcl 14321	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ₀)
9		simplr 766	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 𝑉 ∈ Fin)
10		hashge1 14354	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 1 ≤ (♯‘𝑉))
11	10	ad2ant2l 743	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 1 ≤ (♯‘𝑉))
12		nn0re 12485	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (♯‘𝑉) ∈ ℝ)
13		3re 12296	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
14		lenlt 11296	. . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . 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 14458	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))
26		vex 3472	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
27		eqid 2726	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
28	4, 27	1to3vfriendship 30043	. . . . . . . . . 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 687	. . . . . . . . 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 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 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 395 ∨ w3o 1083 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ∖ cdif 3940 ∅c0 4317 {csn 4623 {cpr 4625 {ctp 4627 class class class wbr 5141 ‘cfv 6537 Fincfn 8941 ℝcr 11111 1c1 11113 < clt 11252 ≤ cle 11253 3c3 12272 ℕ₀cn0 12476 ♯chash 14295 Vtxcvtx 28764 Edgcedg 28815 FriendGraph cfrgr 30020

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-3o 8469 df-oadd 8471 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12981 df-xadd 13099 df-ico 13336 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-reps 14725 df-csh 14745 df-s2 14805 df-s3 14806 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-dvds 16205 df-gcd 16443 df-prm 16616 df-phi 16708 df-vtx 28766 df-iedg 28767 df-edg 28816 df-uhgr 28826 df-ushgr 28827 df-upgr 28850 df-umgr 28851 df-uspgr 28918 df-usgr 28919 df-fusgr 29082 df-nbgr 29098 df-vtxdg 29232 df-rgr 29323 df-rusgr 29324 df-wlks 29365 df-wlkson 29366 df-trls 29458 df-trlson 29459 df-pths 29482 df-spths 29483 df-pthson 29484 df-spthson 29485 df-wwlks 29593 df-wwlksn 29594 df-wwlksnon 29595 df-wspthsn 29596 df-wspthsnon 29597 df-clwwlk 29744 df-clwwlkn 29787 df-clwwlknon 29850 df-conngr 29949 df-frgr 30021

This theorem is referenced by: (None)

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