Theorem nb3grprlem1 26282

Description: Lemma 1 for nb3grpr 26284. (Contributed by Alexander van der Vekens, 15-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 ) )

Assertion

Ref	Expression
nb3grprlem1	|- ( ph -> ( ( G NeighbVtx A ) = { B , C } <-> ( { A , B } e. E /\ { A , C } e. E ) ) )

Proof of Theorem nb3grprlem1

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nb3grpr.s	. . . . . . 7 |- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )
2		prid1g 4295	. . . . . . . 8 |- ( B e. Y -> B e. { B , C } )
3	2	3ad2ant2 1083	. . . . . . 7 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> B e. { B , C } )
4	1, 3	syl 17	. . . . . 6 |- ( ph -> B e. { B , C } )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> B e. { B , C } )
6		eleq2 2690	. . . . . . 7 |- ( { B , C } = ( G NeighbVtx A ) -> ( B e. { B , C } <-> B e. ( G NeighbVtx A ) ) )
7	6	eqcoms 2630	. . . . . 6 |- ( ( G NeighbVtx A ) = { B , C } -> ( B e. { B , C } <-> B e. ( G NeighbVtx A ) ) )
8	7	adantl 482	. . . . 5 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> ( B e. { B , C } <-> B e. ( G NeighbVtx A ) ) )
9	5, 8	mpbid 222	. . . 4 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> B e. ( G NeighbVtx A ) )
10		nb3grpr.g	. . . . . 6 |- ( ph -> G e. USGraph )
11		nb3grpr.e	. . . . . . . 8 |- E = (Edg ` G )
12	11	nbusgreledg 26249	. . . . . . 7 |- ( G e. USGraph -> ( B e. ( G NeighbVtx A ) <-> { B , A } e. E ) )
13		prcom 4267	. . . . . . . . 9 |- { B , A } = { A , B }
14	13	a1i 11	. . . . . . . 8 |- ( G e. USGraph -> { B , A } = { A , B } )
15	14	eleq1d 2686	. . . . . . 7 |- ( G e. USGraph -> ( { B , A } e. E <-> { A , B } e. E ) )
16	12, 15	bitrd 268	. . . . . 6 |- ( G e. USGraph -> ( B e. ( G NeighbVtx A ) <-> { A , B } e. E ) )
17	10, 16	syl 17	. . . . 5 |- ( ph -> ( B e. ( G NeighbVtx A ) <-> { A , B } e. E ) )
18	17	adantr 481	. . . 4 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> ( B e. ( G NeighbVtx A ) <-> { A , B } e. E ) )
19	9, 18	mpbid 222	. . 3 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> { A , B } e. E )
20		prid2g 4296	. . . . . . . 8 |- ( C e. Z -> C e. { B , C } )
21	20	3ad2ant3 1084	. . . . . . 7 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> C e. { B , C } )
22	1, 21	syl 17	. . . . . 6 |- ( ph -> C e. { B , C } )
23	22	adantr 481	. . . . 5 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> C e. { B , C } )
24		eleq2 2690	. . . . . . 7 |- ( { B , C } = ( G NeighbVtx A ) -> ( C e. { B , C } <-> C e. ( G NeighbVtx A ) ) )
25	24	eqcoms 2630	. . . . . 6 |- ( ( G NeighbVtx A ) = { B , C } -> ( C e. { B , C } <-> C e. ( G NeighbVtx A ) ) )
26	25	adantl 482	. . . . 5 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> ( C e. { B , C } <-> C e. ( G NeighbVtx A ) ) )
27	23, 26	mpbid 222	. . . 4 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> C e. ( G NeighbVtx A ) )
28	11	nbusgreledg 26249	. . . . . . 7 |- ( G e. USGraph -> ( C e. ( G NeighbVtx A ) <-> { C , A } e. E ) )
29		prcom 4267	. . . . . . . . 9 |- { C , A } = { A , C }
30	29	a1i 11	. . . . . . . 8 |- ( G e. USGraph -> { C , A } = { A , C } )
31	30	eleq1d 2686	. . . . . . 7 |- ( G e. USGraph -> ( { C , A } e. E <-> { A , C } e. E ) )
32	28, 31	bitrd 268	. . . . . 6 |- ( G e. USGraph -> ( C e. ( G NeighbVtx A ) <-> { A , C } e. E ) )
33	10, 32	syl 17	. . . . 5 |- ( ph -> ( C e. ( G NeighbVtx A ) <-> { A , C } e. E ) )
34	33	adantr 481	. . . 4 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> ( C e. ( G NeighbVtx A ) <-> { A , C } e. E ) )
35	27, 34	mpbid 222	. . 3 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> { A , C } e. E )
36	19, 35	jca 554	. 2 |- ( ( ph /\ ( G NeighbVtx A ) = { B , C } ) -> ( { A , B } e. E /\ { A , C } e. E ) )
37		nb3grpr.v	. . . . . 6 |- V = (Vtx ` G )
38	37, 11	nbusgr 26245	. . . . 5 |- ( G e. USGraph -> ( G NeighbVtx A ) = { v e. V | { A , v } e. E } )
39	10, 38	syl 17	. . . 4 |- ( ph -> ( G NeighbVtx A ) = { v e. V | { A , v } e. E } )
40	39	adantr 481	. . 3 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( G NeighbVtx A ) = { v e. V | { A , v } e. E } )
41		nb3grpr.t	. . . . . . . . . 10 |- ( ph -> V = { A , B , C } )
42		eleq2 2690	. . . . . . . . . 10 |- ( V = { A , B , C } -> ( v e. V <-> v e. { A , B , C } ) )
43	41, 42	syl 17	. . . . . . . . 9 |- ( ph -> ( v e. V <-> v e. { A , B , C } ) )
44	43	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v e. V <-> v e. { A , B , C } ) )
45		vex 3203	. . . . . . . . . . 11 |- v e. _V
46	45	eltp 4230	. . . . . . . . . 10 |- ( v e. { A , B , C } <-> ( v = A \/ v = B \/ v = C ) )
47	11	usgredgne 26098	. . . . . . . . . . . . . . . 16 |- ( ( G e. USGraph /\ { A , v } e. E ) -> A =/= v )
48		df-ne 2795	. . . . . . . . . . . . . . . . 17 |- ( A =/= v <-> -. A = v )
49		pm2.24 121	. . . . . . . . . . . . . . . . . . 19 |- ( A = v -> ( -. A = v -> ( v = B \/ v = C ) ) )
50	49	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 |- ( v = A -> ( -. A = v -> ( v = B \/ v = C ) ) )
51	50	com12 32	. . . . . . . . . . . . . . . . 17 |- ( -. A = v -> ( v = A -> ( v = B \/ v = C ) ) )
52	48, 51	sylbi 207	. . . . . . . . . . . . . . . 16 |- ( A =/= v -> ( v = A -> ( v = B \/ v = C ) ) )
53	47, 52	syl 17	. . . . . . . . . . . . . . 15 |- ( ( G e. USGraph /\ { A , v } e. E ) -> ( v = A -> ( v = B \/ v = C ) ) )
54	53	ex 450	. . . . . . . . . . . . . 14 |- ( G e. USGraph -> ( { A , v } e. E -> ( v = A -> ( v = B \/ v = C ) ) ) )
55	10, 54	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( { A , v } e. E -> ( v = A -> ( v = B \/ v = C ) ) ) )
56	55	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( { A , v } e. E -> ( v = A -> ( v = B \/ v = C ) ) ) )
57	56	com3r 87	. . . . . . . . . . 11 |- ( v = A -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( { A , v } e. E -> ( v = B \/ v = C ) ) ) )
58		orc 400	. . . . . . . . . . . 12 |- ( v = B -> ( v = B \/ v = C ) )
59	58	2a1d 26	. . . . . . . . . . 11 |- ( v = B -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( { A , v } e. E -> ( v = B \/ v = C ) ) ) )
60		olc 399	. . . . . . . . . . . 12 |- ( v = C -> ( v = B \/ v = C ) )
61	60	2a1d 26	. . . . . . . . . . 11 |- ( v = C -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( { A , v } e. E -> ( v = B \/ v = C ) ) ) )
62	57, 59, 61	3jaoi 1391	. . . . . . . . . 10 |- ( ( v = A \/ v = B \/ v = C ) -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( { A , v } e. E -> ( v = B \/ v = C ) ) ) )
63	46, 62	sylbi 207	. . . . . . . . 9 |- ( v e. { A , B , C } -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( { A , v } e. E -> ( v = B \/ v = C ) ) ) )
64	63	com12 32	. . . . . . . 8 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v e. { A , B , C } -> ( { A , v } e. E -> ( v = B \/ v = C ) ) ) )
65	44, 64	sylbid 230	. . . . . . 7 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v e. V -> ( { A , v } e. E -> ( v = B \/ v = C ) ) ) )
66	65	impd 447	. . . . . 6 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( ( v e. V /\ { A , v } e. E ) -> ( v = B \/ v = C ) ) )
67		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- B = B
68	67	3mix2i 1234	. . . . . . . . . . . . . . . . 17 |- ( B = A \/ B = B \/ B = C )
69	1	simp2d 1074	. . . . . . . . . . . . . . . . . 18 |- ( ph -> B e. Y )
70		eltpg 4227	. . . . . . . . . . . . . . . . . 18 |- ( B e. Y -> ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) ) )
71	69, 70	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) ) )
72	68, 71	mpbiri 248	. . . . . . . . . . . . . . . 16 |- ( ph -> B e. { A , B , C } )
73	72	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ v = B ) -> B e. { A , B , C } )
74		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( v = B -> ( v e. { A , B , C } <-> B e. { A , B , C } ) )
75	74	bicomd 213	. . . . . . . . . . . . . . . 16 |- ( v = B -> ( B e. { A , B , C } <-> v e. { A , B , C } ) )
76	75	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ v = B ) -> ( B e. { A , B , C } <-> v e. { A , B , C } ) )
77	73, 76	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ph /\ v = B ) -> v e. { A , B , C } )
78	42	bicomd 213	. . . . . . . . . . . . . . . 16 |- ( V = { A , B , C } -> ( v e. { A , B , C } <-> v e. V ) )
79	41, 78	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( v e. { A , B , C } <-> v e. V ) )
80	79	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ v = B ) -> ( v e. { A , B , C } <-> v e. V ) )
81	77, 80	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ph /\ v = B ) -> v e. V )
82	81	ex 450	. . . . . . . . . . . 12 |- ( ph -> ( v = B -> v e. V ) )
83	82	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v = B -> v e. V ) )
84	83	impcom 446	. . . . . . . . . 10 |- ( ( v = B /\ ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) ) -> v e. V )
85		preq2 4269	. . . . . . . . . . . . . . 15 |- ( B = v -> { A , B } = { A , v } )
86	85	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( B = v -> ( { A , B } e. E <-> { A , v } e. E ) )
87	86	eqcoms 2630	. . . . . . . . . . . . 13 |- ( v = B -> ( { A , B } e. E <-> { A , v } e. E ) )
88	87	biimpcd 239	. . . . . . . . . . . 12 |- ( { A , B } e. E -> ( v = B -> { A , v } e. E ) )
89	88	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v = B -> { A , v } e. E ) )
90	89	impcom 446	. . . . . . . . . 10 |- ( ( v = B /\ ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) ) -> { A , v } e. E )
91	84, 90	jca 554	. . . . . . . . 9 |- ( ( v = B /\ ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) ) -> ( v e. V /\ { A , v } e. E ) )
92	91	ex 450	. . . . . . . 8 |- ( v = B -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v e. V /\ { A , v } e. E ) ) )
93		tpid3g 4305	. . . . . . . . . . . . . . . . . 18 |- ( C e. Z -> C e. { A , B , C } )
94	93	3ad2ant3 1084	. . . . . . . . . . . . . . . . 17 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> C e. { A , B , C } )
95	1, 94	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> C e. { A , B , C } )
96	95	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ v = C ) -> C e. { A , B , C } )
97		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( v = C -> ( v e. { A , B , C } <-> C e. { A , B , C } ) )
98	97	bicomd 213	. . . . . . . . . . . . . . . 16 |- ( v = C -> ( C e. { A , B , C } <-> v e. { A , B , C } ) )
99	98	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ v = C ) -> ( C e. { A , B , C } <-> v e. { A , B , C } ) )
100	96, 99	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ph /\ v = C ) -> v e. { A , B , C } )
101	79	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ v = C ) -> ( v e. { A , B , C } <-> v e. V ) )
102	100, 101	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ph /\ v = C ) -> v e. V )
103	102	ex 450	. . . . . . . . . . . 12 |- ( ph -> ( v = C -> v e. V ) )
104	103	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v = C -> v e. V ) )
105	104	impcom 446	. . . . . . . . . 10 |- ( ( v = C /\ ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) ) -> v e. V )
106		preq2 4269	. . . . . . . . . . . . . . 15 |- ( C = v -> { A , C } = { A , v } )
107	106	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( C = v -> ( { A , C } e. E <-> { A , v } e. E ) )
108	107	eqcoms 2630	. . . . . . . . . . . . 13 |- ( v = C -> ( { A , C } e. E <-> { A , v } e. E ) )
109	108	biimpcd 239	. . . . . . . . . . . 12 |- ( { A , C } e. E -> ( v = C -> { A , v } e. E ) )
110	109	ad2antll 765	. . . . . . . . . . 11 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v = C -> { A , v } e. E ) )
111	110	impcom 446	. . . . . . . . . 10 |- ( ( v = C /\ ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) ) -> { A , v } e. E )
112	105, 111	jca 554	. . . . . . . . 9 |- ( ( v = C /\ ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) ) -> ( v e. V /\ { A , v } e. E ) )
113	112	ex 450	. . . . . . . 8 |- ( v = C -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v e. V /\ { A , v } e. E ) ) )
114	92, 113	jaoi 394	. . . . . . 7 |- ( ( v = B \/ v = C ) -> ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( v e. V /\ { A , v } e. E ) ) )
115	114	com12 32	. . . . . 6 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( ( v = B \/ v = C ) -> ( v e. V /\ { A , v } e. E ) ) )
116	66, 115	impbid 202	. . . . 5 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( ( v e. V /\ { A , v } e. E ) <-> ( v = B \/ v = C ) ) )
117	116	abbidv 2741	. . . 4 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> { v | ( v e. V /\ { A , v } e. E ) } = { v | ( v = B \/ v = C ) } )
118		df-rab 2921	. . . 4 |- { v e. V | { A , v } e. E } = { v | ( v e. V /\ { A , v } e. E ) }
119		dfpr2 4195	. . . 4 |- { B , C } = { v | ( v = B \/ v = C ) }
120	117, 118, 119	3eqtr4g 2681	. . 3 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> { v e. V | { A , v } e. E } = { B , C } )
121	40, 120	eqtrd 2656	. 2 |- ( ( ph /\ ( { A , B } e. E /\ { A , C } e. E ) ) -> ( G NeighbVtx A ) = { B , C } )
122	36, 121	impbida 877	1 |- ( ph -> ( ( G NeighbVtx A ) = { B , C } <-> ( { A , B } e. E /\ { A , C } e. E ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   {crab 2916   {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:  nb3grpr  26284  nb3grpr2  26285  nb3gr2nb  26286