Theorem uvtxanbgrvtx 26293

Description: A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)

Hypothesis

Ref	Expression
uvtxael.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
uvtxanbgrvtx	|- ( N e. (UnivVtx ` G ) -> A. v e. ( V \ { N } ) N e. ( G NeighbVtx v ) )

Distinct variable groups:   v, G   v, N   v, V

Proof of Theorem uvtxanbgrvtx

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxael.v	. . . 4 |- V = (Vtx ` G )
2	1	vtxnbuvtx 26291	. . 3 |- ( N e. (UnivVtx ` G ) -> A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) )
3		eleq1 2689	. . . . . . 7 |- ( n = v -> ( n e. ( G NeighbVtx N ) <-> v e. ( G NeighbVtx N ) ) )
4	3	rspcva 3307	. . . . . 6 |- ( ( v e. ( V \ { N } ) /\ A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) -> v e. ( G NeighbVtx N ) )
5		elfvex 6221	. . . . . . 7 |- ( N e. (UnivVtx ` G ) -> G e. _V )
6		nbgrsym 26265	. . . . . . 7 |- ( G e. _V -> ( v e. ( G NeighbVtx N ) <-> N e. ( G NeighbVtx v ) ) )
7	5, 6	syl 17	. . . . . 6 |- ( N e. (UnivVtx ` G ) -> ( v e. ( G NeighbVtx N ) <-> N e. ( G NeighbVtx v ) ) )
8	4, 7	syl5ibcom 235	. . . . 5 |- ( ( v e. ( V \ { N } ) /\ A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) -> ( N e. (UnivVtx ` G ) -> N e. ( G NeighbVtx v ) ) )
9	8	expcom 451	. . . 4 |- ( A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) -> ( v e. ( V \ { N } ) -> ( N e. (UnivVtx ` G ) -> N e. ( G NeighbVtx v ) ) ) )
10	9	com23 86	. . 3 |- ( A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) -> ( N e. (UnivVtx ` G ) -> ( v e. ( V \ { N } ) -> N e. ( G NeighbVtx v ) ) ) )
11	2, 10	mpcom 38	. 2 |- ( N e. (UnivVtx ` G ) -> ( v e. ( V \ { N } ) -> N e. ( G NeighbVtx v ) ) )
12	11	ralrimiv 2965	1 |- ( N e. (UnivVtx ` G ) -> A. v e. ( V \ { N } ) N e. ( G NeighbVtx v ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874   NeighbVtx cnbgr 26224  UnivVtxcuvtxa 26225

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-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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-nbgr 26228  df-uvtxa 26230

This theorem is referenced by: (None)