Theorem cplgr3v 26331

Description: A pseudograph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.)

Hypotheses

Ref	Expression
cplgr3v.e	|- E = (Edg ` G )
cplgr3v.t	|- (Vtx ` G ) = { A , B , C }

Assertion

Ref	Expression
cplgr3v	|- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( G e. ComplGraph <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) )

Proof of Theorem cplgr3v

Dummy variables n v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cplgr3v.t	. . . . 5 |- (Vtx ` G ) = { A , B , C }
2	1	eqcomi 2631	. . . 4 |- { A , B , C } = (Vtx ` G )
3	2	iscplgrnb 26312	. . 3 |- ( G e. UPGraph -> ( G e. ComplGraph <-> A. v e. { A , B , C } A. n e. ( { A , B , C } \ { v } ) n e. ( G NeighbVtx v ) ) )
4	3	3ad2ant2 1083	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( G e. ComplGraph <-> A. v e. { A , B , C } A. n e. ( { A , B , C } \ { v } ) n e. ( G NeighbVtx v ) ) )
5		sneq 4187	. . . . . 6 |- ( v = A -> { v } = { A } )
6	5	difeq2d 3728	. . . . 5 |- ( v = A -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { A } ) )
7		tprot 4284	. . . . . . . 8 |- { A , B , C } = { B , C , A }
8	7	difeq1i 3724	. . . . . . 7 |- ( { A , B , C } \ { A } ) = ( { B , C , A } \ { A } )
9		necom 2847	. . . . . . . . 9 |- ( A =/= B <-> B =/= A )
10		necom 2847	. . . . . . . . 9 |- ( A =/= C <-> C =/= A )
11		diftpsn3 4332	. . . . . . . . 9 |- ( ( B =/= A /\ C =/= A ) -> ( { B , C , A } \ { A } ) = { B , C } )
12	9, 10, 11	syl2anb 496	. . . . . . . 8 |- ( ( A =/= B /\ A =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
13	12	3adant3 1081	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
14	8, 13	syl5eq 2668	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = { B , C } )
15	14	3ad2ant3 1084	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( { A , B , C } \ { A } ) = { B , C } )
16	6, 15	sylan9eqr 2678	. . . 4 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = A ) -> ( { A , B , C } \ { v } ) = { B , C } )
17		oveq2 6658	. . . . . 6 |- ( v = A -> ( G NeighbVtx v ) = ( G NeighbVtx A ) )
18	17	eleq2d 2687	. . . . 5 |- ( v = A -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx A ) ) )
19	18	adantl 482	. . . 4 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = A ) -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx A ) ) )
20	16, 19	raleqbidv 3152	. . 3 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = A ) -> ( A. n e. ( { A , B , C } \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. { B , C } n e. ( G NeighbVtx A ) ) )
21		sneq 4187	. . . . . 6 |- ( v = B -> { v } = { B } )
22	21	difeq2d 3728	. . . . 5 |- ( v = B -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { B } ) )
23		tprot 4284	. . . . . . . . 9 |- { C , A , B } = { A , B , C }
24	23	eqcomi 2631	. . . . . . . 8 |- { A , B , C } = { C , A , B }
25	24	difeq1i 3724	. . . . . . 7 |- ( { A , B , C } \ { B } ) = ( { C , A , B } \ { B } )
26		necom 2847	. . . . . . . . . . . 12 |- ( B =/= C <-> C =/= B )
27	26	biimpi 206	. . . . . . . . . . 11 |- ( B =/= C -> C =/= B )
28	27	anim2i 593	. . . . . . . . . 10 |- ( ( A =/= B /\ B =/= C ) -> ( A =/= B /\ C =/= B ) )
29	28	ancomd 467	. . . . . . . . 9 |- ( ( A =/= B /\ B =/= C ) -> ( C =/= B /\ A =/= B ) )
30		diftpsn3 4332	. . . . . . . . 9 |- ( ( C =/= B /\ A =/= B ) -> ( { C , A , B } \ { B } ) = { C , A } )
31	29, 30	syl 17	. . . . . . . 8 |- ( ( A =/= B /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
32	31	3adant2 1080	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
33	25, 32	syl5eq 2668	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = { C , A } )
34	33	3ad2ant3 1084	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( { A , B , C } \ { B } ) = { C , A } )
35	22, 34	sylan9eqr 2678	. . . 4 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = B ) -> ( { A , B , C } \ { v } ) = { C , A } )
36		oveq2 6658	. . . . . 6 |- ( v = B -> ( G NeighbVtx v ) = ( G NeighbVtx B ) )
37	36	eleq2d 2687	. . . . 5 |- ( v = B -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx B ) ) )
38	37	adantl 482	. . . 4 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = B ) -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx B ) ) )
39	35, 38	raleqbidv 3152	. . 3 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = B ) -> ( A. n e. ( { A , B , C } \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. { C , A } n e. ( G NeighbVtx B ) ) )
40		sneq 4187	. . . . . 6 |- ( v = C -> { v } = { C } )
41	40	difeq2d 3728	. . . . 5 |- ( v = C -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { C } ) )
42		diftpsn3 4332	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
43	42	3adant1 1079	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
44	43	3ad2ant3 1084	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( { A , B , C } \ { C } ) = { A , B } )
45	41, 44	sylan9eqr 2678	. . . 4 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = C ) -> ( { A , B , C } \ { v } ) = { A , B } )
46		oveq2 6658	. . . . . 6 |- ( v = C -> ( G NeighbVtx v ) = ( G NeighbVtx C ) )
47	46	eleq2d 2687	. . . . 5 |- ( v = C -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx C ) ) )
48	47	adantl 482	. . . 4 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = C ) -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx C ) ) )
49	45, 48	raleqbidv 3152	. . 3 |- ( ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ v = C ) -> ( A. n e. ( { A , B , C } \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. { A , B } n e. ( G NeighbVtx C ) ) )
50		simp1 1061	. . . 4 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. X )
51	50	3ad2ant1 1082	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> A e. X )
52		simp2 1062	. . . 4 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> B e. Y )
53	52	3ad2ant1 1082	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> B e. Y )
54		simp3 1063	. . . 4 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> C e. Z )
55	54	3ad2ant1 1082	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> C e. Z )
56	20, 39, 49, 51, 53, 55	raltpd 4315	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( A. v e. { A , B , C } A. n e. ( { A , B , C } \ { v } ) n e. ( G NeighbVtx v ) <-> ( A. n e. { B , C } n e. ( G NeighbVtx A ) /\ A. n e. { C , A } n e. ( G NeighbVtx B ) /\ A. n e. { A , B } n e. ( G NeighbVtx C ) ) ) )
57		eleq1 2689	. . . . . . 7 |- ( n = B -> ( n e. ( G NeighbVtx A ) <-> B e. ( G NeighbVtx A ) ) )
58		eleq1 2689	. . . . . . 7 |- ( n = C -> ( n e. ( G NeighbVtx A ) <-> C e. ( G NeighbVtx A ) ) )
59	57, 58	ralprg 4234	. . . . . 6 |- ( ( B e. Y /\ C e. Z ) -> ( A. n e. { B , C } n e. ( G NeighbVtx A ) <-> ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx A ) ) ) )
60	59	3adant1 1079	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( A. n e. { B , C } n e. ( G NeighbVtx A ) <-> ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx A ) ) ) )
61		eleq1 2689	. . . . . . . 8 |- ( n = C -> ( n e. ( G NeighbVtx B ) <-> C e. ( G NeighbVtx B ) ) )
62		eleq1 2689	. . . . . . . 8 |- ( n = A -> ( n e. ( G NeighbVtx B ) <-> A e. ( G NeighbVtx B ) ) )
63	61, 62	ralprg 4234	. . . . . . 7 |- ( ( C e. Z /\ A e. X ) -> ( A. n e. { C , A } n e. ( G NeighbVtx B ) <-> ( C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx B ) ) ) )
64	63	ancoms 469	. . . . . 6 |- ( ( A e. X /\ C e. Z ) -> ( A. n e. { C , A } n e. ( G NeighbVtx B ) <-> ( C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx B ) ) ) )
65	64	3adant2 1080	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( A. n e. { C , A } n e. ( G NeighbVtx B ) <-> ( C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx B ) ) ) )
66		eleq1 2689	. . . . . . 7 |- ( n = A -> ( n e. ( G NeighbVtx C ) <-> A e. ( G NeighbVtx C ) ) )
67		eleq1 2689	. . . . . . 7 |- ( n = B -> ( n e. ( G NeighbVtx C ) <-> B e. ( G NeighbVtx C ) ) )
68	66, 67	ralprg 4234	. . . . . 6 |- ( ( A e. X /\ B e. Y ) -> ( A. n e. { A , B } n e. ( G NeighbVtx C ) <-> ( A e. ( G NeighbVtx C ) /\ B e. ( G NeighbVtx C ) ) ) )
69	68	3adant3 1081	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( A. n e. { A , B } n e. ( G NeighbVtx C ) <-> ( A e. ( G NeighbVtx C ) /\ B e. ( G NeighbVtx C ) ) ) )
70	60, 65, 69	3anbi123d 1399	. . . 4 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( A. n e. { B , C } n e. ( G NeighbVtx A ) /\ A. n e. { C , A } n e. ( G NeighbVtx B ) /\ A. n e. { A , B } n e. ( G NeighbVtx C ) ) <-> ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx B ) ) /\ ( A e. ( G NeighbVtx C ) /\ B e. ( G NeighbVtx C ) ) ) ) )
71	70	3ad2ant1 1082	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( A. n e. { B , C } n e. ( G NeighbVtx A ) /\ A. n e. { C , A } n e. ( G NeighbVtx B ) /\ A. n e. { A , B } n e. ( G NeighbVtx C ) ) <-> ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx B ) ) /\ ( A e. ( G NeighbVtx C ) /\ B e. ( G NeighbVtx C ) ) ) ) )
72		3an6 1409	. . . 4 |- ( ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx B ) ) /\ ( A e. ( G NeighbVtx C ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx C ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) )
73	72	a1i 11	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx B ) ) /\ ( A e. ( G NeighbVtx C ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx C ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) ) )
74		nbgrsym 26265	. . . . . . 7 |- ( G e. UPGraph -> ( B e. ( G NeighbVtx A ) <-> A e. ( G NeighbVtx B ) ) )
75		nbgrsym 26265	. . . . . . 7 |- ( G e. UPGraph -> ( C e. ( G NeighbVtx B ) <-> B e. ( G NeighbVtx C ) ) )
76		nbgrsym 26265	. . . . . . 7 |- ( G e. UPGraph -> ( A e. ( G NeighbVtx C ) <-> C e. ( G NeighbVtx A ) ) )
77	74, 75, 76	3anbi123d 1399	. . . . . 6 |- ( G e. UPGraph -> ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx C ) ) <-> ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) ) )
78	77	3ad2ant2 1083	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx C ) ) <-> ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) ) )
79	78	anbi1d 741	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx C ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) ) )
80		3anrot 1043	. . . . . . . 8 |- ( ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) <-> ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) )
81	80	bicomi 214	. . . . . . 7 |- ( ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) <-> ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) )
82	81	anbi1i 731	. . . . . 6 |- ( ( ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) )
83		anidm 676	. . . . . 6 |- ( ( ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) )
84	82, 83	bitri 264	. . . . 5 |- ( ( ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) )
85	84	a1i 11	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) )
86		tpid3g 4305	. . . . . . . . 9 |- ( A e. X -> A e. { B , C , A } )
87	86, 7	syl6eleqr 2712	. . . . . . . 8 |- ( A e. X -> A e. { A , B , C } )
88		tpid3g 4305	. . . . . . . . 9 |- ( B e. Y -> B e. { C , A , B } )
89	88, 24	syl6eleqr 2712	. . . . . . . 8 |- ( B e. Y -> B e. { A , B , C } )
90		tpid3g 4305	. . . . . . . 8 |- ( C e. Z -> C e. { A , B , C } )
91	87, 89, 90	3anim123i 1247	. . . . . . 7 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( A e. { A , B , C } /\ B e. { A , B , C } /\ C e. { A , B , C } ) )
92		df-3an 1039	. . . . . . 7 |- ( ( A e. { A , B , C } /\ B e. { A , B , C } /\ C e. { A , B , C } ) <-> ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) )
93	91, 92	sylib 208	. . . . . 6 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) )
94		simpr 477	. . . . . . . . . . . 12 |- ( ( A e. { A , B , C } /\ B e. { A , B , C } ) -> B e. { A , B , C } )
95	94	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) -> B e. { A , B , C } )
96	95	anim1i 592	. . . . . . . . . 10 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph ) -> ( B e. { A , B , C } /\ G e. UPGraph ) )
97	96	ancomd 467	. . . . . . . . 9 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph ) -> ( G e. UPGraph /\ B e. { A , B , C } ) )
98	97	3adant3 1081	. . . . . . . 8 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( G e. UPGraph /\ B e. { A , B , C } ) )
99		simpll 790	. . . . . . . . . 10 |- ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) -> A e. { A , B , C } )
100		simp1 1061	. . . . . . . . . 10 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> A =/= B )
101	99, 100	anim12i 590	. . . . . . . . 9 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( A e. { A , B , C } /\ A =/= B ) )
102	101	3adant2 1080	. . . . . . . 8 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( A e. { A , B , C } /\ A =/= B ) )
103		cplgr3v.e	. . . . . . . . 9 |- E = (Edg ` G )
104	2, 103	nbupgrel 26241	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ B e. { A , B , C } ) /\ ( A e. { A , B , C } /\ A =/= B ) ) -> ( A e. ( G NeighbVtx B ) <-> { A , B } e. E ) )
105	98, 102, 104	syl2anc 693	. . . . . . 7 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( A e. ( G NeighbVtx B ) <-> { A , B } e. E ) )
106		simpr 477	. . . . . . . . . . 11 |- ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) -> C e. { A , B , C } )
107	106	anim1i 592	. . . . . . . . . 10 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph ) -> ( C e. { A , B , C } /\ G e. UPGraph ) )
108	107	ancomd 467	. . . . . . . . 9 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph ) -> ( G e. UPGraph /\ C e. { A , B , C } ) )
109	108	3adant3 1081	. . . . . . . 8 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( G e. UPGraph /\ C e. { A , B , C } ) )
110		simp3 1063	. . . . . . . . . 10 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> B =/= C )
111	95, 110	anim12i 590	. . . . . . . . 9 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( B e. { A , B , C } /\ B =/= C ) )
112	111	3adant2 1080	. . . . . . . 8 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( B e. { A , B , C } /\ B =/= C ) )
113	2, 103	nbupgrel 26241	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ C e. { A , B , C } ) /\ ( B e. { A , B , C } /\ B =/= C ) ) -> ( B e. ( G NeighbVtx C ) <-> { B , C } e. E ) )
114	109, 112, 113	syl2anc 693	. . . . . . 7 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( B e. ( G NeighbVtx C ) <-> { B , C } e. E ) )
115	99	anim1i 592	. . . . . . . . . 10 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph ) -> ( A e. { A , B , C } /\ G e. UPGraph ) )
116	115	ancomd 467	. . . . . . . . 9 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph ) -> ( G e. UPGraph /\ A e. { A , B , C } ) )
117	116	3adant3 1081	. . . . . . . 8 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( G e. UPGraph /\ A e. { A , B , C } ) )
118		simp2 1062	. . . . . . . . . . 11 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> A =/= C )
119	118	necomd 2849	. . . . . . . . . 10 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> C =/= A )
120	106, 119	anim12i 590	. . . . . . . . 9 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( C e. { A , B , C } /\ C =/= A ) )
121	120	3adant2 1080	. . . . . . . 8 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( C e. { A , B , C } /\ C =/= A ) )
122	2, 103	nbupgrel 26241	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ A e. { A , B , C } ) /\ ( C e. { A , B , C } /\ C =/= A ) ) -> ( C e. ( G NeighbVtx A ) <-> { C , A } e. E ) )
123	117, 121, 122	syl2anc 693	. . . . . . 7 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( C e. ( G NeighbVtx A ) <-> { C , A } e. E ) )
124	105, 114, 123	3anbi123d 1399	. . . . . 6 |- ( ( ( ( A e. { A , B , C } /\ B e. { A , B , C } ) /\ C e. { A , B , C } ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) )
125	93, 124	syl3an1 1359	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) /\ C e. ( G NeighbVtx A ) ) <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) )
126	80, 125	syl5bb 272	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) )
127	79, 85, 126	3bitrd 294	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( B e. ( G NeighbVtx A ) /\ C e. ( G NeighbVtx B ) /\ A e. ( G NeighbVtx C ) ) /\ ( C e. ( G NeighbVtx A ) /\ A e. ( G NeighbVtx B ) /\ B e. ( G NeighbVtx C ) ) ) <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) )
128	71, 73, 127	3bitrd 294	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( A. n e. { B , C } n e. ( G NeighbVtx A ) /\ A. n e. { C , A } n e. ( G NeighbVtx B ) /\ A. n e. { A , B } n e. ( G NeighbVtx C ) ) <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) )
129	4, 56, 128	3bitrd 294	1 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ G e. UPGraph /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( G e. ComplGraph <-> ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   {csn 4177   {cpr 4179   {ctp 4181   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   UPGraph cupgr 25975   NeighbVtx cnbgr 26224  ComplGraphccplgr 26226

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-nbgr 26228  df-uvtxa 26230  df-cplgr 26231

This theorem is referenced by:  cusgr3vnbpr  26332