friendship - Metamath Proof Explorer

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Theorem friendship 30378

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 30377	. . . 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 14297	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ₀)
9		simplr 768	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 𝑉 ∈ Fin)
10		hashge1 14330	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 1 ≤ (♯‘𝑉))
11	10	ad2ant2l 746	. . . . . . . . . . 11 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin) ∧ (¬ 3 < (♯‘𝑉) ∧ 𝑉 ≠ ∅)) → 1 ≤ (♯‘𝑉))
12		nn0re 12427	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (♯‘𝑉) ∈ ℝ)
13		3re 12242	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
14		lenlt 11228	. . . . . . . . . . . . . . . . 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 14433	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 1 ≤ (♯‘𝑉) ∧ (♯‘𝑉) ≤ 3) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))
26		vex 3448	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
27		eqid 2729	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
28	4, 27	1to3vfriendship 30260	. . . . . . . . . 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 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3444 ∖ cdif 3908 ∅c0 4292 {csn 4585 {cpr 4587 {ctp 4589 class class class wbr 5102 ‘cfv 6499 Fincfn 8895 ℝcr 11043 1c1 11045 < clt 11184 ≤ cle 11185 3c3 12218 ℕ₀cn0 12418 ♯chash 14271 Vtxcvtx 28976 Edgcedg 29027 FriendGraph cfrgr 30237

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-ac2 10392 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-3o 8413 df-oadd 8415 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-ac 10045 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-xnn0 12492 df-z 12506 df-uz 12770 df-rp 12928 df-xadd 13049 df-ico 13288 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-hash 14272 df-word 14455 df-lsw 14504 df-concat 14512 df-s1 14537 df-substr 14582 df-pfx 14612 df-reps 14710 df-csh 14730 df-s2 14790 df-s3 14791 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-sum 15629 df-dvds 16199 df-gcd 16441 df-prm 16618 df-phi 16712 df-vtx 28978 df-iedg 28979 df-edg 29028 df-uhgr 29038 df-ushgr 29039 df-upgr 29062 df-umgr 29063 df-uspgr 29130 df-usgr 29131 df-fusgr 29297 df-nbgr 29313 df-vtxdg 29447 df-rgr 29538 df-rusgr 29539 df-wlks 29580 df-wlkson 29581 df-trls 29671 df-trlson 29672 df-pths 29694 df-spths 29695 df-pthson 29696 df-spthson 29697 df-wwlks 29810 df-wwlksn 29811 df-wwlksnon 29812 df-wspthsn 29813 df-wspthsnon 29814 df-clwwlk 29961 df-clwwlkn 30004 df-clwwlknon 30067 df-conngr 30166 df-frgr 30238

This theorem is referenced by: (None)

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