Theorem nbgrssvwo2 26261

Description: The neighbors of a vertex are a subset of all vertices except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)

Hypothesis

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

Assertion

Ref	Expression
nbgrssvwo2	|- ( ( G e. W /\ M e/ ( G NeighbVtx N ) ) -> ( G NeighbVtx N ) C_ ( V \ { M , N } ) )

Proof of Theorem nbgrssvwo2

Step	Hyp	Ref	Expression
1		nbgrssovtx.v	. . . . 5 |- V = (Vtx ` G )
2	1	nbgrssovtx 26260	. . . 4 |- ( G e. W -> ( G NeighbVtx N ) C_ ( V \ { N } ) )
3		df-nel 2898	. . . . . 6 |- ( M e/ ( G NeighbVtx N ) <-> -. M e. ( G NeighbVtx N ) )
4		disjsn 4246	. . . . . 6 |- ( ( ( G NeighbVtx N ) i^i { M } ) = (/) <-> -. M e. ( G NeighbVtx N ) )
5	3, 4	sylbb2 228	. . . . 5 |- ( M e/ ( G NeighbVtx N ) -> ( ( G NeighbVtx N ) i^i { M } ) = (/) )
6		reldisj 4020	. . . . 5 |- ( ( G NeighbVtx N ) C_ ( V \ { N } ) -> ( ( ( G NeighbVtx N ) i^i { M } ) = (/) <-> ( G NeighbVtx N ) C_ ( ( V \ { N } ) \ { M } ) ) )
7	5, 6	syl5ib 234	. . . 4 |- ( ( G NeighbVtx N ) C_ ( V \ { N } ) -> ( M e/ ( G NeighbVtx N ) -> ( G NeighbVtx N ) C_ ( ( V \ { N } ) \ { M } ) ) )
8	2, 7	syl 17	. . 3 |- ( G e. W -> ( M e/ ( G NeighbVtx N ) -> ( G NeighbVtx N ) C_ ( ( V \ { N } ) \ { M } ) ) )
9	8	imp 445	. 2 |- ( ( G e. W /\ M e/ ( G NeighbVtx N ) ) -> ( G NeighbVtx N ) C_ ( ( V \ { N } ) \ { M } ) )
10		prcom 4267	. . . 4 |- { M , N } = { N , M }
11	10	difeq2i 3725	. . 3 |- ( V \ { M , N } ) = ( V \ { N , M } )
12		difpr 4334	. . 3 |- ( V \ { N , M } ) = ( ( V \ { N } ) \ { M } )
13	11, 12	eqtri 2644	. 2 |- ( V \ { M , N } ) = ( ( V \ { N } ) \ { M } )
14	9, 13	syl6sseqr 3652	1 |- ( ( G e. W /\ M e/ ( G NeighbVtx N ) ) -> ( G NeighbVtx N ) C_ ( V \ { M , N } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874   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-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

This theorem is referenced by:  usgrnbssvwo2  26264  nbfusgrlevtxm2  26280