Theorem nbgrcl 26233

Description: If a class has at least one neighbor, it must be a vertex. (Contributed by AV, 6-Jun-2021.)

Assertion

Ref	Expression
nbgrcl	|- ( N e. ( G NeighbVtx X ) -> X e. (Vtx ` G ) )

Proof of Theorem nbgrcl

Dummy variables g e n v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nbgr 26228	. . 3 |- NeighbVtx = ( g e. _V , v e. (Vtx ` g ) |-> { n e. ( (Vtx ` g ) \ { v } ) | E. e e. (Edg ` g ) { v , n } C_ e } )
2	1	mpt2xeldm 7337	. 2 |- ( N e. ( G NeighbVtx X ) -> ( G e. _V /\ X e. [_ G / g ]_ (Vtx ` g ) ) )
3		csbfv 6233	. . . 4 |- [_ G / g ]_ (Vtx ` g ) = (Vtx ` G )
4	3	eleq2i 2693	. . 3 |- ( X e. [_ G / g ]_ (Vtx ` g ) <-> X e. (Vtx ` G ) )
5	4	biimpi 206	. 2 |- ( X e. [_ G / g ]_ (Vtx ` g ) -> X e. (Vtx ` G ) )
6	2, 5	simpl2im 658	1 |- ( N e. ( G NeighbVtx X ) -> X e. (Vtx ` G ) )

Syntax hints:   -> wi 4   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   [_csb 3533   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   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-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:  nbgrel  26238  frgrnbnb  27157