Theorem nbgr0vtxlem 26251

Description: Lemma for nbgr0vtx 26252 and nbgr0edg 26253. (Contributed by AV, 15-Nov-2020.)

Hypothesis

Ref	Expression
nbgr0vtxlem.v	|- ( ph -> A. n e. ( (Vtx ` G ) \ { K } ) -. E. e e. (Edg ` G ) { K , n } C_ e )

Assertion

Ref	Expression
nbgr0vtxlem	|- ( ph -> ( G NeighbVtx K ) = (/) )

Distinct variable groups:   e, G, n   e, K, n

Allowed substitution hints:   ph( e, n)

Proof of Theorem nbgr0vtxlem

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . . . . . . . 9 |- (Edg ` G ) = (Edg ` G )
3	1, 2	nbgrval 26234	. . . . . . . 8 |- ( K e. (Vtx ` G ) -> ( G NeighbVtx K ) = { n e. ( (Vtx ` G ) \ { K } ) | E. e e. (Edg ` G ) { K , n } C_ e } )
4	3	adantr 481	. . . . . . 7 |- ( ( K e. (Vtx ` G ) /\ ph ) -> ( G NeighbVtx K ) = { n e. ( (Vtx ` G ) \ { K } ) | E. e e. (Edg ` G ) { K , n } C_ e } )
5	4	adantl 482	. . . . . 6 |- ( ( ( G e. _V /\ K e. _V ) /\ ( K e. (Vtx ` G ) /\ ph ) ) -> ( G NeighbVtx K ) = { n e. ( (Vtx ` G ) \ { K } ) | E. e e. (Edg ` G ) { K , n } C_ e } )
6		nbgr0vtxlem.v	. . . . . . . 8 |- ( ph -> A. n e. ( (Vtx ` G ) \ { K } ) -. E. e e. (Edg ` G ) { K , n } C_ e )
7	6	ad2antll 765	. . . . . . 7 |- ( ( ( G e. _V /\ K e. _V ) /\ ( K e. (Vtx ` G ) /\ ph ) ) -> A. n e. ( (Vtx ` G ) \ { K } ) -. E. e e. (Edg ` G ) { K , n } C_ e )
8		rabeq0 3957	. . . . . . 7 |- ( { n e. ( (Vtx ` G ) \ { K } ) | E. e e. (Edg ` G ) { K , n } C_ e } = (/) <-> A. n e. ( (Vtx ` G ) \ { K } ) -. E. e e. (Edg ` G ) { K , n } C_ e )
9	7, 8	sylibr 224	. . . . . 6 |- ( ( ( G e. _V /\ K e. _V ) /\ ( K e. (Vtx ` G ) /\ ph ) ) -> { n e. ( (Vtx ` G ) \ { K } ) | E. e e. (Edg ` G ) { K , n } C_ e } = (/) )
10	5, 9	eqtrd 2656	. . . . 5 |- ( ( ( G e. _V /\ K e. _V ) /\ ( K e. (Vtx ` G ) /\ ph ) ) -> ( G NeighbVtx K ) = (/) )
11	10	expcom 451	. . . 4 |- ( ( K e. (Vtx ` G ) /\ ph ) -> ( ( G e. _V /\ K e. _V ) -> ( G NeighbVtx K ) = (/) ) )
12	11	ex 450	. . 3 |- ( K e. (Vtx ` G ) -> ( ph -> ( ( G e. _V /\ K e. _V ) -> ( G NeighbVtx K ) = (/) ) ) )
13	12	com23 86	. 2 |- ( K e. (Vtx ` G ) -> ( ( G e. _V /\ K e. _V ) -> ( ph -> ( G NeighbVtx K ) = (/) ) ) )
14		df-nel 2898	. . . 4 |- ( K e/ (Vtx ` G ) <-> -. K e. (Vtx ` G ) )
15	1	nbgrnvtx0 26237	. . . 4 |- ( K e/ (Vtx ` G ) -> ( G NeighbVtx K ) = (/) )
16	14, 15	sylbir 225	. . 3 |- ( -. K e. (Vtx ` G ) -> ( G NeighbVtx K ) = (/) )
17	16	a1d 25	. 2 |- ( -. K e. (Vtx ` G ) -> ( ph -> ( G NeighbVtx K ) = (/) ) )
18		nbgrprc0 26229	. . 3 |- ( -. ( G e. _V /\ K e. _V ) -> ( G NeighbVtx K ) = (/) )
19	18	a1d 25	. 2 |- ( -. ( G e. _V /\ K e. _V ) -> ( ph -> ( G NeighbVtx K ) = (/) ) )
20	13, 17, 19	pm2.61nii 178	1 |- ( ph -> ( G NeighbVtx K ) = (/) )

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   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:  nbgr0vtx  26252  nbgr0edg  26253  nbgr1vtx  26254