Theorem nbuhgr2vtx1edgb 39584

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 5889	. . . . 5 |- (Vtx ` G ) e. _V
3	1, 2	eqeltri 2545	. . . 4 |- V e. _V
4		hash2prb 12674	. . . 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 468	. . . . . . . . . . . 12 |- ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) -> ( a e. V /\ b e. V ) )
7	6	ancomd 458	. . . . . . . . . . 11 |- ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) -> ( b e. V /\ a e. V ) )
8	7	ad2antrr 740	. . . . . . . . . 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 2698	. . . . . . . . . . . 12 |- ( a =/= b -> b =/= a )
11	10	adantr 472	. . . . . . . . . . 11 |- ( ( a =/= b /\ V = { a , b } ) -> b =/= a )
12	11	ad2antlr 741	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ ( a e. V /\ b e. V ) ) /\ ( a =/= b /\ V = { a , b } ) ) /\ { a , b } e. E ) -> b =/= a )
13		prcom 4041	. . . . . . . . . . . . . 14 |- { a , b } = { b , a }
14	13	eleq1i 2540	. . . . . . . . . . . . 13 |- ( { a , b } e. E <-> { b , a } e. E )
15	14	biimpi 199	. . . . . . . . . . . 12 |- ( { a , b } e. E -> { b , a } e. E )
16		sseq2 3440	. . . . . . . . . . . . 13 |- ( e = { b , a } -> ( { a , b } C_ e <-> { a , b } C_ { b , a } ) )
17	16	adantl 473	. . . . . . . . . . . 12 |- ( ( { a , b } e. E /\ e = { b , a } ) -> ( { a , b } C_ e <-> { a , b } C_ { b , a } ) )
18	13	eqimssi 3472	. . . . . . . . . . . . 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 3143	. . . . . . . . . . 11 |- ( { a , b } e. E -> E. e e. E { a , b } C_ e )
21	20	adantl 473	. . . . . . . . . 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 39574	. . . . . . . . . . 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 744	. . . . . . . . . 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 1213	. . . . . . . . 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 740	. . . . . . . . . 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 778	. . . . . . . . . 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 3440	. . . . . . . . . . . . 13 |- ( e = { a , b } -> ( { b , a } C_ e <-> { b , a } C_ { a , b } ) )
30	29	adantl 473	. . . . . . . . . . . 12 |- ( ( { a , b } e. E /\ e = { a , b } ) -> ( { b , a } C_ e <-> { b , a } C_ { a , b } ) )
31		prcom 4041	. . . . . . . . . . . . . 14 |- { b , a } = { a , b }
32	31	eqimssi 3472	. . . . . . . . . . . . 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 3143	. . . . . . . . . . 11 |- ( { a , b } e. E -> E. e e. E { b , a } C_ e )
35	34	adantl 473	. . . . . . . . . 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 39574	. . . . . . . . . . 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 744	. . . . . . . . . 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 1213	. . . . . . . . 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 541	. . . . . . . 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 441	. . . . . . 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 39583	. . . . . . . . . . . 12 |- ( ( G e. UHGraph /\ V = { a , b } /\ a e. ( G NeighbVtx b ) ) -> { a , b } e. E )
42	41	3exp 1230	. . . . . . . . . . 11 |- ( G e. UHGraph -> ( V = { a , b } -> ( a e. ( G NeighbVtx b ) -> { a , b } e. E ) ) )
43	42	adantr 472	. . . . . . . . . 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 474	. . . . . . . . 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 436	. . . . . . . 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 474	. . . . . . 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 195	. . . . . 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 2537	. . . . . . . . 9 |- ( V = { a , b } -> ( V e. E <-> { a , b } e. E ) )
49	48	adantl 473	. . . . . . . 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 3533	. . . . . . . . . . 11 |- ( V = { a , b } -> ( V \ { v } ) = ( { a , b } \ { v } ) )
52	51	raleqdv 2979	. . . . . . . . . 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 2987	. . . . . . . . 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 3034	. . . . . . . . . . 11 |- a e. _V
55		vex 3034	. . . . . . . . . . 11 |- b e. _V
56		sneq 3969	. . . . . . . . . . . . 13 |- ( v = a -> { v } = { a } )
57	56	difeq2d 3540	. . . . . . . . . . . 12 |- ( v = a -> ( { a , b } \ { v } ) = ( { a , b } \ { a } ) )
58		oveq2 6316	. . . . . . . . . . . . 13 |- ( v = a -> ( G NeighbVtx v ) = ( G NeighbVtx a ) )
59	58	eleq2d 2534	. . . . . . . . . . . 12 |- ( v = a -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx a ) ) )
60	57, 59	raleqbidv 2987	. . . . . . . . . . 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 3969	. . . . . . . . . . . . 13 |- ( v = b -> { v } = { b } )
62	61	difeq2d 3540	. . . . . . . . . . . 12 |- ( v = b -> ( { a , b } \ { v } ) = ( { a , b } \ { b } ) )
63		oveq2 6316	. . . . . . . . . . . . 13 |- ( v = b -> ( G NeighbVtx v ) = ( G NeighbVtx b ) )
64	63	eleq2d 2534	. . . . . . . . . . . 12 |- ( v = b -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx b ) ) )
65	62, 64	raleqbidv 2987	. . . . . . . . . . 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 4016	. . . . . . . . . 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 4099	. . . . . . . . . . . . 13 |- ( a =/= b -> ( { a , b } \ { a } ) = { b } )
68	67	raleqdv 2979	. . . . . . . . . . . 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 2537	. . . . . . . . . . . . 13 |- ( n = b -> ( n e. ( G NeighbVtx a ) <-> b e. ( G NeighbVtx a ) ) )
70	55, 69	ralsn 4001	. . . . . . . . . . . 12 |- ( A. n e. { b } n e. ( G NeighbVtx a ) <-> b e. ( G NeighbVtx a ) )
71	68, 70	syl6bb 269	. . . . . . . . . . 11 |- ( a =/= b -> ( A. n e. ( { a , b } \ { a } ) n e. ( G NeighbVtx a ) <-> b e. ( G NeighbVtx a ) ) )
72		difprsn2 4100	. . . . . . . . . . . . 13 |- ( a =/= b -> ( { a , b } \ { b } ) = { a } )
73	72	raleqdv 2979	. . . . . . . . . . . 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 2537	. . . . . . . . . . . . 13 |- ( n = a -> ( n e. ( G NeighbVtx b ) <-> a e. ( G NeighbVtx b ) ) )
75	54, 74	ralsn 4001	. . . . . . . . . . . 12 |- ( A. n e. { a } n e. ( G NeighbVtx b ) <-> a e. ( G NeighbVtx b ) )
76	73, 75	syl6bb 269	. . . . . . . . . . 11 |- ( a =/= b -> ( A. n e. ( { a , b } \ { b } ) n e. ( G NeighbVtx b ) <-> a e. ( G NeighbVtx b ) ) )
77	71, 76	anbi12d 725	. . . . . . . . . 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 265	. . . . . . . . 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 715	. . . . . . . 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 328	. . . . . . 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 473	. . . . . 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 240	. . . . 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 441	. . . 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 2878	. . 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 225	. 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 436	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   \ cdif 3387   C_ wss 3390   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308   2c2 10681   #chash 12553  Vtxcvtx 39251   UHGraph cuhgr 39300  Edgcedga 39371   NeighbVtx cnbgr 39561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-uhgr 39302  df-edga 39372  df-nbgr 39565

This theorem is referenced by:  uvtx2vtx1edgb  39636