Theorem frcond3 30250

Description: The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhood of a vertex and the neighborhood of a second, different vertex have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 30-Dec-2021.)

Hypotheses

Ref	Expression
frcond1.v	⊢ 𝑉 = (Vtx‘𝐺)
frcond1.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frcond3	⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐸   𝑥,𝐺   𝑥,𝑉

Proof of Theorem frcond3

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frcond1.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		frcond1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	frcond1 30247	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
4	3	imp 406	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
5		ssrab2 4055	. . . . . . . . . 10 ⊢ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉
6		sseq1 3984	. . . . . . . . . 10 ⊢ ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
7	5, 6	mpbii 233	. . . . . . . . 9 ⊢ ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
8		vex 3463	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9	8	snss 4761	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑉 ↔ {𝑥} ⊆ 𝑉)
10	7, 9	sylibr 234	. . . . . . . 8 ⊢ ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → 𝑥 ∈ 𝑉)
11	10	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → 𝑥 ∈ 𝑉)
12		frgrusgr 30242	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
13	1, 2	nbusgr 29328	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸})
14	1, 2	nbusgr 29328	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐶) = {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸})
15	13, 14	ineq12d 4196	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
16	12, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ FriendGraph → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
18	17	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
19		inrab 4291	. . . . . . . . 9 ⊢ ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}) = {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)}
20	18, 19	eqtrdi 2786	. . . . . . . 8 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)})
21		prcom 4708	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝑏} = {𝑏, 𝐶}
22	21	eleq1i 2825	. . . . . . . . . . . . 13 ⊢ ({𝐶, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝐶} ∈ 𝐸)
23	22	anbi2i 623	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
24		prex 5407	. . . . . . . . . . . . 13 ⊢ {𝐴, 𝑏} ∈ V
25		prex 5407	. . . . . . . . . . . . 13 ⊢ {𝑏, 𝐶} ∈ V
26	24, 25	prss 4796	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
27	23, 26	bitri 275	. . . . . . . . . . 11 ⊢ (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
28	27	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ 𝑏 ∈ 𝑉) → (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
29	28	rabbidva 3422	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
31		simpr 484	. . . . . . . 8 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
32	20, 30, 31	3eqtrd 2774	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
33	11, 32	jca 511	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → (𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
34	33	ex 412	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → (𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
35	34	eximdv 1917	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → (∃𝑥{𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
36		reusn 4703	. . . 4 ⊢ (∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃𝑥{𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
37		df-rex 3061	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
38	35, 36, 37	3imtr4g 296	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → (∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
39	4, 38	mpd 15	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
40	39	ex 412	1 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060  ∃!wreu 3357  {crab 3415   ∩ cin 3925   ⊆ wss 3926  {csn 4601  {cpr 4603  ‘cfv 6531  (class class class)co 7405  Vtxcvtx 28975  Edgcedg 29026  USGraphcusgr 29128   NeighbVtx cnbgr 29311   FriendGraph cfrgr 30239

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-n0 12502  df-xnn0 12575  df-z 12589  df-uz 12853  df-fz 13525  df-hash 14349  df-edg 29027  df-upgr 29061  df-umgr 29062  df-usgr 29130  df-nbgr 29312  df-frgr 30240

This theorem is referenced by:  frcond4  30251  frgrncvvdeqlem3  30282