Theorem nb3grpr 26284

Description: The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
nb3grpr.v	|- V = (Vtx ` G )
nb3grpr.e	|- E = (Edg ` G )
nb3grpr.g	|- ( ph -> G e. USGraph )
nb3grpr.t	|- ( ph -> V = { A , B , C } )
nb3grpr.s	|- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )
nb3grpr.n	|- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )

Assertion

Ref	Expression
nb3grpr	|- ( ph -> ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> A. x e. V E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } ) )

Distinct variable groups:   x, A, y, z   x, B, y, z   x, C, y, z   y, E   x, G, y, z   x, V, y, z   ph, y

Allowed substitution hints:   ph( x, z)   E( x, z)   X( x, y, z)   Y( x, y, z)   Z( x, y, z)

Proof of Theorem nb3grpr

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 |- ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) -> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) )
2		prcom 4267	. . . . . . . . . 10 |- { A , B } = { B , A }
3	2	eleq1i 2692	. . . . . . . . 9 |- ( { A , B } e. E <-> { B , A } e. E )
4		prcom 4267	. . . . . . . . . 10 |- { B , C } = { C , B }
5	4	eleq1i 2692	. . . . . . . . 9 |- ( { B , C } e. E <-> { C , B } e. E )
6		prcom 4267	. . . . . . . . . 10 |- { C , A } = { A , C }
7	6	eleq1i 2692	. . . . . . . . 9 |- ( { C , A } e. E <-> { A , C } e. E )
8	3, 5, 7	3anbi123i 1251	. . . . . . . 8 |- ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( { B , A } e. E /\ { C , B } e. E /\ { A , C } e. E ) )
9		3anrot 1043	. . . . . . . 8 |- ( ( { A , C } e. E /\ { B , A } e. E /\ { C , B } e. E ) <-> ( { B , A } e. E /\ { C , B } e. E /\ { A , C } e. E ) )
10	8, 9	bitr4i 267	. . . . . . 7 |- ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( { A , C } e. E /\ { B , A } e. E /\ { C , B } e. E ) )
11	10	a1i 11	. . . . . 6 |- ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) -> ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( { A , C } e. E /\ { B , A } e. E /\ { C , B } e. E ) ) )
12	1, 11	biadan2 674	. . . . 5 |- ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) /\ ( { A , C } e. E /\ { B , A } e. E /\ { C , B } e. E ) ) )
13		an6 1408	. . . . 5 |- ( ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) /\ ( { A , C } e. E /\ { B , A } e. E /\ { C , B } e. E ) ) <-> ( ( { A , B } e. E /\ { A , C } e. E ) /\ ( { B , C } e. E /\ { B , A } e. E ) /\ ( { C , A } e. E /\ { C , B } e. E ) ) )
14	12, 13	bitri 264	. . . 4 |- ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( ( { A , B } e. E /\ { A , C } e. E ) /\ ( { B , C } e. E /\ { B , A } e. E ) /\ ( { C , A } e. E /\ { C , B } e. E ) ) )
15	14	a1i 11	. . 3 |- ( ph -> ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( ( { A , B } e. E /\ { A , C } e. E ) /\ ( { B , C } e. E /\ { B , A } e. E ) /\ ( { C , A } e. E /\ { C , B } e. E ) ) ) )
16		nb3grpr.v	. . . . 5 |- V = (Vtx ` G )
17		nb3grpr.e	. . . . 5 |- E = (Edg ` G )
18		nb3grpr.g	. . . . 5 |- ( ph -> G e. USGraph )
19		nb3grpr.t	. . . . 5 |- ( ph -> V = { A , B , C } )
20		nb3grpr.s	. . . . 5 |- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )
21	16, 17, 18, 19, 20	nb3grprlem1 26282	. . . 4 |- ( ph -> ( ( G NeighbVtx A ) = { B , C } <-> ( { A , B } e. E /\ { A , C } e. E ) ) )
22		tprot 4284	. . . . . 6 |- { A , B , C } = { B , C , A }
23	19, 22	syl6eq 2672	. . . . 5 |- ( ph -> V = { B , C , A } )
24		3anrot 1043	. . . . . 6 |- ( ( A e. X /\ B e. Y /\ C e. Z ) <-> ( B e. Y /\ C e. Z /\ A e. X ) )
25	20, 24	sylib 208	. . . . 5 |- ( ph -> ( B e. Y /\ C e. Z /\ A e. X ) )
26	16, 17, 18, 23, 25	nb3grprlem1 26282	. . . 4 |- ( ph -> ( ( G NeighbVtx B ) = { C , A } <-> ( { B , C } e. E /\ { B , A } e. E ) ) )
27		tprot 4284	. . . . . 6 |- { C , A , B } = { A , B , C }
28	19, 27	syl6eqr 2674	. . . . 5 |- ( ph -> V = { C , A , B } )
29		3anrot 1043	. . . . . 6 |- ( ( C e. Z /\ A e. X /\ B e. Y ) <-> ( A e. X /\ B e. Y /\ C e. Z ) )
30	20, 29	sylibr 224	. . . . 5 |- ( ph -> ( C e. Z /\ A e. X /\ B e. Y ) )
31	16, 17, 18, 28, 30	nb3grprlem1 26282	. . . 4 |- ( ph -> ( ( G NeighbVtx C ) = { A , B } <-> ( { C , A } e. E /\ { C , B } e. E ) ) )
32	21, 26, 31	3anbi123d 1399	. . 3 |- ( ph -> ( ( ( G NeighbVtx A ) = { B , C } /\ ( G NeighbVtx B ) = { C , A } /\ ( G NeighbVtx C ) = { A , B } ) <-> ( ( { A , B } e. E /\ { A , C } e. E ) /\ ( { B , C } e. E /\ { B , A } e. E ) /\ ( { C , A } e. E /\ { C , B } e. E ) ) ) )
33		nb3grpr.n	. . . . 5 |- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )
34	16, 17, 18, 19, 20, 33	nb3grprlem2 26283	. . . 4 |- ( ph -> ( ( G NeighbVtx A ) = { B , C } <-> E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx A ) = { y , z } ) )
35		necom 2847	. . . . . . . 8 |- ( A =/= B <-> B =/= A )
36		necom 2847	. . . . . . . 8 |- ( A =/= C <-> C =/= A )
37		biid 251	. . . . . . . 8 |- ( B =/= C <-> B =/= C )
38	35, 36, 37	3anbi123i 1251	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) <-> ( B =/= A /\ C =/= A /\ B =/= C ) )
39		3anrot 1043	. . . . . . 7 |- ( ( B =/= C /\ B =/= A /\ C =/= A ) <-> ( B =/= A /\ C =/= A /\ B =/= C ) )
40	38, 39	bitr4i 267	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) <-> ( B =/= C /\ B =/= A /\ C =/= A ) )
41	33, 40	sylib 208	. . . . 5 |- ( ph -> ( B =/= C /\ B =/= A /\ C =/= A ) )
42	16, 17, 18, 23, 25, 41	nb3grprlem2 26283	. . . 4 |- ( ph -> ( ( G NeighbVtx B ) = { C , A } <-> E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx B ) = { y , z } ) )
43		3anrot 1043	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) <-> ( A =/= C /\ B =/= C /\ A =/= B ) )
44		necom 2847	. . . . . . . 8 |- ( B =/= C <-> C =/= B )
45		biid 251	. . . . . . . 8 |- ( A =/= B <-> A =/= B )
46	36, 44, 45	3anbi123i 1251	. . . . . . 7 |- ( ( A =/= C /\ B =/= C /\ A =/= B ) <-> ( C =/= A /\ C =/= B /\ A =/= B ) )
47	43, 46	bitri 264	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) <-> ( C =/= A /\ C =/= B /\ A =/= B ) )
48	33, 47	sylib 208	. . . . 5 |- ( ph -> ( C =/= A /\ C =/= B /\ A =/= B ) )
49	16, 17, 18, 28, 30, 48	nb3grprlem2 26283	. . . 4 |- ( ph -> ( ( G NeighbVtx C ) = { A , B } <-> E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx C ) = { y , z } ) )
50	34, 42, 49	3anbi123d 1399	. . 3 |- ( ph -> ( ( ( G NeighbVtx A ) = { B , C } /\ ( G NeighbVtx B ) = { C , A } /\ ( G NeighbVtx C ) = { A , B } ) <-> ( E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx A ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx B ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx C ) = { y , z } ) ) )
51	15, 32, 50	3bitr2d 296	. 2 |- ( ph -> ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx A ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx B ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx C ) = { y , z } ) ) )
52		oveq2 6658	. . . . . 6 |- ( x = A -> ( G NeighbVtx x ) = ( G NeighbVtx A ) )
53	52	eqeq1d 2624	. . . . 5 |- ( x = A -> ( ( G NeighbVtx x ) = { y , z } <-> ( G NeighbVtx A ) = { y , z } ) )
54	53	2rexbidv 3057	. . . 4 |- ( x = A -> ( E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx A ) = { y , z } ) )
55		oveq2 6658	. . . . . 6 |- ( x = B -> ( G NeighbVtx x ) = ( G NeighbVtx B ) )
56	55	eqeq1d 2624	. . . . 5 |- ( x = B -> ( ( G NeighbVtx x ) = { y , z } <-> ( G NeighbVtx B ) = { y , z } ) )
57	56	2rexbidv 3057	. . . 4 |- ( x = B -> ( E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx B ) = { y , z } ) )
58		oveq2 6658	. . . . . 6 |- ( x = C -> ( G NeighbVtx x ) = ( G NeighbVtx C ) )
59	58	eqeq1d 2624	. . . . 5 |- ( x = C -> ( ( G NeighbVtx x ) = { y , z } <-> ( G NeighbVtx C ) = { y , z } ) )
60	59	2rexbidv 3057	. . . 4 |- ( x = C -> ( E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx C ) = { y , z } ) )
61	54, 57, 60	raltpg 4236	. . 3 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( A. x e. { A , B , C } E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> ( E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx A ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx B ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx C ) = { y , z } ) ) )
62	20, 61	syl 17	. 2 |- ( ph -> ( A. x e. { A , B , C } E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> ( E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx A ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx B ) = { y , z } /\ E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx C ) = { y , z } ) ) )
63		raleq 3138	. . . 4 |- ( V = { A , B , C } -> ( A. x e. V E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> A. x e. { A , B , C } E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } ) )
64	63	bicomd 213	. . 3 |- ( V = { A , B , C } -> ( A. x e. { A , B , C } E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> A. x e. V E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } ) )
65	19, 64	syl 17	. 2 |- ( ph -> ( A. x e. { A , B , C } E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } <-> A. x e. V E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } ) )
66	51, 62, 65	3bitr2d 296	1 |- ( ph -> ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> A. x e. V E. y e. V E. z e. ( V \ { y } ) ( G NeighbVtx x ) = { y , z } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   {csn 4177   {cpr 4179   {ctp 4181   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228

This theorem is referenced by:  cusgr3vnbpr  26332