Theorem nbuhgr2vtx1edgb 26248

Description: If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.)

Hypotheses

Ref	Expression
nbgr2vtx1edg.v	|- V = (Vtx ` G )
nbgr2vtx1edg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
nbuhgr2vtx1edgb	|- ( ( G e. UHGraph /\ ( # ` V ) = 2 ) -> ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )

Distinct variable groups:   n, E   n, G, v   n, V, v

Allowed substitution hint:   E( v)

Proof of Theorem nbuhgr2vtx1edgb

Dummy variables a b e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgr2vtx1edg.v	. . . . 5 |- V = (Vtx ` G )
2		fvex 6201	. . . . 5 |- (Vtx ` G ) e. _V
3	1, 2	eqeltri 2697	. . . 4 |- V e. _V
4		hash2prb 13254	. . . 4 |- ( V e. _V -> ( ( # ` V ) = 2 <-> E. a e. V E. b e. V ( a =/= b /\ V = { a , b } ) ) )
5	3, 4	ax-mp 5	. . 3 |- ( ( # ` V ) = 2 <-> E. a e. V E. b e. V ( a =/= b /\ V = { a , b } ) )
6		simpr 477	. . . . . . . . . . . 12 |- ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) -> ( a e. V /\ b e. V ) )
7	6	ancomd 467	. . . . . . . . . . 11 |- ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) -> ( b e. V /\ a e. V ) )
8	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> ( b e. V /\ a e. V ) )
9		id 22	. . . . . . . . . . . . 13 |- ( a =/= b -> a =/= b )
10	9	necomd 2849	. . . . . . . . . . . 12 |- ( a =/= b -> b =/= a )
11	10	adantr 481	. . . . . . . . . . 11 |- ( ( a =/= b /\ V = { a , b } ) -> b =/= a )
12	11	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> b =/= a )
13		prcom 4267	. . . . . . . . . . . . . 14 |- { a , b } = { b , a }
14	13	eleq1i 2692	. . . . . . . . . . . . 13 |- ( { a , b } e. E <-> { b , a } e. E )
15	14	biimpi 206	. . . . . . . . . . . 12 |- ( { a , b } e. E -> { b , a } e. E )
16		sseq2 3627	. . . . . . . . . . . . 13 |- ( e = { b , a } -> ( { a , b } C_ e <-> { a , b } C_ { b , a } ) )
17	16	adantl 482	. . . . . . . . . . . 12 |- ( ( { a , b } e. E /\ e = { b , a } ) -> ( { a , b } C_ e <-> { a , b } C_ { b , a } ) )
18	13	eqimssi 3659	. . . . . . . . . . . . 13 |- { a , b } C_ { b , a }
19	18	a1i 11	. . . . . . . . . . . 12 |- ( { a , b } e. E -> { a , b } C_ { b , a } )
20	15, 17, 19	rspcedvd 3317	. . . . . . . . . . 11 |- ( { a , b } e. E -> E. e e. E { a , b } C_ e )
21	20	adantl 482	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> E. e e. E { a , b } C_ e )
22		nbgr2vtx1edg.e	. . . . . . . . . . . 12 |- E = (Edg ` G )
23	1, 22	nbgrel 26238	. . . . . . . . . . 11 |- ( G e. UHGraph -> ( b e. ( G NeighbVtx a ) <-> ( ( b e. V /\ a e. V ) /\ b =/= a /\ E. e e. E { a , b } C_ e ) ) )
24	23	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> ( b e. ( G NeighbVtx a ) <-> ( ( b e. V /\ a e. V ) /\ b =/= a /\ E. e e. E { a , b } C_ e ) ) )
25	8, 12, 21, 24	mpbir3and 1245	. . . . . . . . 9 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> b e. ( G NeighbVtx a ) )
26	6	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> ( a e. V /\ b e. V ) )
27		simplrl 800	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> a =/= b )
28		id 22	. . . . . . . . . . . 12 |- ( { a , b } e. E -> { a , b } e. E )
29		sseq2 3627	. . . . . . . . . . . . 13 |- ( e = { a , b } -> ( { b , a } C_ e <-> { b , a } C_ { a , b } ) )
30	29	adantl 482	. . . . . . . . . . . 12 |- ( ( { a , b } e. E /\ e = { a , b } ) -> ( { b , a } C_ e <-> { b , a } C_ { a , b } ) )
31		prcom 4267	. . . . . . . . . . . . . 14 |- { b , a } = { a , b }
32	31	eqimssi 3659	. . . . . . . . . . . . 13 |- { b , a } C_ { a , b }
33	32	a1i 11	. . . . . . . . . . . 12 |- ( { a , b } e. E -> { b , a } C_ { a , b } )
34	28, 30, 33	rspcedvd 3317	. . . . . . . . . . 11 |- ( { a , b } e. E -> E. e e. E { b , a } C_ e )
35	34	adantl 482	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> E. e e. E { b , a } C_ e )
36	1, 22	nbgrel 26238	. . . . . . . . . . 11 |- ( G e. UHGraph -> ( a e. ( G NeighbVtx b ) <-> ( ( a e. V /\ b e. V ) /\ a =/= b /\ E. e e. E { b , a } C_ e ) ) )
37	36	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> ( a e. ( G NeighbVtx b ) <-> ( ( a e. V /\ b e. V ) /\ a =/= b /\ E. e e. E { b , a } C_ e ) ) )
38	26, 27, 35, 37	mpbir3and 1245	. . . . . . . . 9 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> a e. ( G NeighbVtx b ) )
39	25, 38	jca 554	. . . . . . . 8 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) )
40	39	ex 450	. . . . . . 7 |- ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) -> ( { a , b } e. E -> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) ) )
41	1, 22	nbuhgr2vtx1edgblem 26247	. . . . . . . . . . . 12 |- ( ( G e. UHGraph /\ V = { a , b } /\ a e. ( G NeighbVtx b ) ) -> { a , b } e. E )
42	41	3exp 1264	. . . . . . . . . . 11 |- ( G e. UHGraph -> ( V = { a , b } -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) ) )
43	42	adantr 481	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) -> ( V = { a , b } -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) ) )
44	43	adantld 483	. . . . . . . . 9 |- ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) -> ( ( a =/= b /\ V = { a , b } ) -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) ) )
45	44	imp 445	. . . . . . . 8 |- ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) )
46	45	adantld 483	. . . . . . 7 |- ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) -> ( ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) -> { a , b } e. E ) )
47	40, 46	impbid 202	. . . . . 6 |- ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) -> ( { a , b } e. E <-> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) ) )
48		eleq1 2689	. . . . . . . . 9 |- ( V = { a , b } -> ( V e. E <-> { a , b } e. E ) )
49	48	adantl 482	. . . . . . . 8 |- ( ( a =/= b /\ V = { a , b } ) -> ( V e. E <-> { a , b } e. E ) )
50		id 22	. . . . . . . . . 10 |- ( V = { a , b } -> V = { a , b } )
51		difeq1 3721	. . . . . . . . . . 11 |- ( V = { a , b } -> ( V \ { v } ) = ( { a , b } \ { v } ) )
52	51	raleqdv 3144	. . . . . . . . . 10 |- ( V = { a , b } -> ( A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. ( { a , b } \ { v } ) n e. ( G NeighbVtx v ) ) )
53	50, 52	raleqbidv 3152	. . . . . . . . 9 |- ( V = { a , b } -> ( A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> A. v e. { a , b } A. n e. ( { a , b } \ { v } ) n e. ( G NeighbVtx v ) ) )
54		vex 3203	. . . . . . . . . . 11 |- a e. _V
55		vex 3203	. . . . . . . . . . 11 |- b e. _V
56		sneq 4187	. . . . . . . . . . . . 13 |- ( v = a -> { v } = { a } )
57	56	difeq2d 3728	. . . . . . . . . . . 12 |- ( v = a -> ( { a , b } \ { v } ) = ( { a , b } \ { a } ) )
58		oveq2 6658	. . . . . . . . . . . . 13 |- ( v = a -> ( G NeighbVtx v ) = ( G NeighbVtx a ) )
59	58	eleq2d 2687	. . . . . . . . . . . 12 |- ( v = a -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx a ) ) )
60	57, 59	raleqbidv 3152	. . . . . . . . . . 11 |- ( v = a -> ( A. n e. ( { a , b } \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. ( { a , b } \ { a } ) n e. ( G NeighbVtx a ) ) )
61		sneq 4187	. . . . . . . . . . . . 13 |- ( v = b -> { v } = { b } )
62	61	difeq2d 3728	. . . . . . . . . . . 12 |- ( v = b -> ( { a , b } \ { v } ) = ( { a , b } \ { b } ) )
63		oveq2 6658	. . . . . . . . . . . . 13 |- ( v = b -> ( G NeighbVtx v ) = ( G NeighbVtx b ) )
64	63	eleq2d 2687	. . . . . . . . . . . 12 |- ( v = b -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx b ) ) )
65	62, 64	raleqbidv 3152	. . . . . . . . . . 11 |- ( v = b -> ( A. n e. ( { a , b } \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. ( { a , b } \ { b } ) n e. ( G NeighbVtx b ) ) )
66	54, 55, 60, 65	ralpr 4238	. . . . . . . . . 10 |- ( A. v e. { a , b } A. n e. ( { a , b } \ { v } ) n e. ( G NeighbVtx v ) <-> ( A. n e. ( { a , b } \ { a } ) n e. ( G NeighbVtx a ) /\ A. n e. ( { a , b } \ { b } ) n e. ( G NeighbVtx b ) ) )
67		difprsn1 4330	. . . . . . . . . . . . 13 |- ( a =/= b -> ( { a , b } \ { a } ) = { b } )
68	67	raleqdv 3144	. . . . . . . . . . . 12 |- ( a =/= b -> ( A. n e. ( { a , b } \ { a } ) n e. ( G NeighbVtx a ) <-> A. n e. { b } n e. ( G NeighbVtx a ) ) )
69		eleq1 2689	. . . . . . . . . . . . 13 |- ( n = b -> ( n e. ( G NeighbVtx a ) <-> b e. ( G NeighbVtx a ) ) )
70	55, 69	ralsn 4222	. . . . . . . . . . . 12 |- ( A. n e. { b } n e. ( G NeighbVtx a ) <-> b e. ( G NeighbVtx a ) )
71	68, 70	syl6bb 276	. . . . . . . . . . 11 |- ( a =/= b -> ( A. n e. ( { a , b } \ { a } ) n e. ( G NeighbVtx a ) <-> b e. ( G NeighbVtx a ) ) )
72		difprsn2 4331	. . . . . . . . . . . . 13 |- ( a =/= b -> ( { a , b } \ { b } ) = { a } )
73	72	raleqdv 3144	. . . . . . . . . . . 12 |- ( a =/= b -> ( A. n e. ( { a , b } \ { b } ) n e. ( G NeighbVtx b ) <-> A. n e. { a } n e. ( G NeighbVtx b ) ) )
74		eleq1 2689	. . . . . . . . . . . . 13 |- ( n = a -> ( n e. ( G NeighbVtx b ) <-> a e. ( G NeighbVtx b ) ) )
75	54, 74	ralsn 4222	. . . . . . . . . . . 12 |- ( A. n e. { a } n e. ( G NeighbVtx b ) <-> a e. ( G NeighbVtx b ) )
76	73, 75	syl6bb 276	. . . . . . . . . . 11 |- ( a =/= b -> ( A. n e. ( { a , b } \ { b } ) n e. ( G NeighbVtx b ) <-> a e. ( G NeighbVtx b ) ) )
77	71, 76	anbi12d 747	. . . . . . . . . 10 |- ( a =/= b -> ( ( A. n e. ( { a , b } \ { a } ) n e. ( G NeighbVtx a ) /\ A. n e. ( { a , b } \ { b } ) n e. ( G NeighbVtx b ) ) <-> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) ) )
78	66, 77	syl5bb 272	. . . . . . . . 9 |- ( a =/= b -> ( A. v e. { a , b } A. n e. ( { a , b } \ { v } ) n e. ( G NeighbVtx v ) <-> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) ) )
79	53, 78	sylan9bbr 737	. . . . . . . 8 |- ( ( a =/= b /\ V = { a , b } ) -> ( A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) ) )
80	49, 79	bibi12d 335	. . . . . . 7 |- ( ( a =/= b /\ V = { a , b } ) -> ( ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) <-> ( { a , b } e. E <-> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) ) ) )
81	80	adantl 482	. . . . . 6 |- ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) -> ( ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) <-> ( { a , b } e. E <-> ( b e. ( G NeighbVtx a ) /\ a e. ( G NeighbVtx b ) ) ) ) )
82	47, 81	mpbird 247	. . . . 5 |- ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) -> ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
83	82	ex 450	. . . 4 |- ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) -> ( ( a =/= b /\ V = { a , b } ) -> ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) ) )
84	83	rexlimdvva 3038	. . 3 |- ( G e. UHGraph -> ( E. a e. V E. b e. V ( a =/= b /\ V = { a , b } ) -> ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) ) )
85	5, 84	syl5bi 232	. 2 |- ( G e. UHGraph -> ( ( # ` V ) = 2 -> ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) ) )
86	85	imp 445	1 |- ( ( G e. UHGraph /\ ( # ` V ) = 2 ) -> ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   2c2 11070   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   UHGraph cuhgr 25951   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-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-nbgr 26228

This theorem is referenced by:  uvtx2vtx1edgb  26300