Theorem uhgrnbgr0nb 26250

Description: A vertex which is not endpoint of an edge has no neighbor in a hypergraph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)

Assertion

Ref	Expression
uhgrnbgr0nb	|- ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) -> ( G NeighbVtx N ) = (/) )

Distinct variable groups:   e, G   e, N

Proof of Theorem uhgrnbgr0nb

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . . . . 6 |- (Edg ` G ) = (Edg ` G )
3	1, 2	nbuhgr 26239	. . . . 5 |- ( ( G e. UHGraph /\ N e. _V ) -> ( G NeighbVtx N ) = { n e. ( (Vtx ` G ) \ { N } ) | E. e e. (Edg ` G ) { N , n } C_ e } )
4	3	adantlr 751	. . . 4 |- ( ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) /\ N e. _V ) -> ( G NeighbVtx N ) = { n e. ( (Vtx ` G ) \ { N } ) | E. e e. (Edg ` G ) { N , n } C_ e } )
5		df-nel 2898	. . . . . . . . . . . . . 14 |- ( N e/ e <-> -. N e. e )
6		prssg 4350	. . . . . . . . . . . . . . . . 17 |- ( ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) -> ( ( N e. e /\ n e. e ) <-> { N , n } C_ e ) )
7		simpl 473	. . . . . . . . . . . . . . . . 17 |- ( ( N e. e /\ n e. e ) -> N e. e )
8	6, 7	syl6bir 244	. . . . . . . . . . . . . . . 16 |- ( ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) -> ( { N , n } C_ e -> N e. e ) )
9	8	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UHGraph /\ ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) ) /\ e e. (Edg ` G ) ) -> ( { N , n } C_ e -> N e. e ) )
10	9	con3d 148	. . . . . . . . . . . . . 14 |- ( ( ( G e. UHGraph /\ ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) ) /\ e e. (Edg ` G ) ) -> ( -. N e. e -> -. { N , n } C_ e ) )
11	5, 10	syl5bi 232	. . . . . . . . . . . . 13 |- ( ( ( G e. UHGraph /\ ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) ) /\ e e. (Edg ` G ) ) -> ( N e/ e -> -. { N , n } C_ e ) )
12	11	ralimdva 2962	. . . . . . . . . . . 12 |- ( ( G e. UHGraph /\ ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) ) -> ( A. e e. (Edg ` G ) N e/ e -> A. e e. (Edg ` G ) -. { N , n } C_ e ) )
13	12	imp 445	. . . . . . . . . . 11 |- ( ( ( G e. UHGraph /\ ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) ) /\ A. e e. (Edg ` G ) N e/ e ) -> A. e e. (Edg ` G ) -. { N , n } C_ e )
14		ralnex 2992	. . . . . . . . . . 11 |- ( A. e e. (Edg ` G ) -. { N , n } C_ e <-> -. E. e e. (Edg ` G ) { N , n } C_ e )
15	13, 14	sylib 208	. . . . . . . . . 10 |- ( ( ( G e. UHGraph /\ ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) ) /\ A. e e. (Edg ` G ) N e/ e ) -> -. E. e e. (Edg ` G ) { N , n } C_ e )
16	15	expcom 451	. . . . . . . . 9 |- ( A. e e. (Edg ` G ) N e/ e -> ( ( G e. UHGraph /\ ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) ) -> -. E. e e. (Edg ` G ) { N , n } C_ e ) )
17	16	expd 452	. . . . . . . 8 |- ( A. e e. (Edg ` G ) N e/ e -> ( G e. UHGraph -> ( ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) -> -. E. e e. (Edg ` G ) { N , n } C_ e ) ) )
18	17	impcom 446	. . . . . . 7 |- ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) -> ( ( N e. _V /\ n e. ( (Vtx ` G ) \ { N } ) ) -> -. E. e e. (Edg ` G ) { N , n } C_ e ) )
19	18	expdimp 453	. . . . . 6 |- ( ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) /\ N e. _V ) -> ( n e. ( (Vtx ` G ) \ { N } ) -> -. E. e e. (Edg ` G ) { N , n } C_ e ) )
20	19	ralrimiv 2965	. . . . 5 |- ( ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) /\ N e. _V ) -> A. n e. ( (Vtx ` G ) \ { N } ) -. E. e e. (Edg ` G ) { N , n } C_ e )
21		rabeq0 3957	. . . . 5 |- ( { n e. ( (Vtx ` G ) \ { N } ) | E. e e. (Edg ` G ) { N , n } C_ e } = (/) <-> A. n e. ( (Vtx ` G ) \ { N } ) -. E. e e. (Edg ` G ) { N , n } C_ e )
22	20, 21	sylibr 224	. . . 4 |- ( ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) /\ N e. _V ) -> { n e. ( (Vtx ` G ) \ { N } ) | E. e e. (Edg ` G ) { N , n } C_ e } = (/) )
23	4, 22	eqtrd 2656	. . 3 |- ( ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) /\ N e. _V ) -> ( G NeighbVtx N ) = (/) )
24	23	expcom 451	. 2 |- ( N e. _V -> ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) -> ( G NeighbVtx N ) = (/) ) )
25		id 22	. . . . 5 |- ( -. N e. _V -> -. N e. _V )
26	25	intnand 962	. . . 4 |- ( -. N e. _V -> -. ( G e. _V /\ N e. _V ) )
27		nbgrprc0 26229	. . . 4 |- ( -. ( G e. _V /\ N e. _V ) -> ( G NeighbVtx N ) = (/) )
28	26, 27	syl 17	. . 3 |- ( -. N e. _V -> ( G NeighbVtx N ) = (/) )
29	28	a1d 25	. 2 |- ( -. N e. _V -> ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) -> ( G NeighbVtx N ) = (/) ) )
30	24, 29	pm2.61i 176	1 |- ( ( G e. UHGraph /\ A. e e. (Edg ` G ) N e/ e ) -> ( G NeighbVtx N ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-edg 25940  df-uhgr 25953  df-nbgr 26228

This theorem is referenced by: (None)