Theorem nbusgrvtxm1 26281

Description: If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
hashnbusgrnn0.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
nbusgrvtxm1	|- ( ( G e. FinUSGraph /\ U e. V ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= U ) -> M e. ( G NeighbVtx U ) ) ) )

Proof of Theorem nbusgrvtxm1

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 |- ( M e. ( G NeighbVtx U ) -> ( ( M e. V /\ M =/= U ) -> M e. ( G NeighbVtx U ) ) )
2	1	2a1d 26	. 2 |- ( M e. ( G NeighbVtx U ) -> ( ( G e. FinUSGraph /\ U e. V ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= U ) -> M e. ( G NeighbVtx U ) ) ) ) )
3		simpr 477	. . . . . . . 8 |- ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) -> ( G e. FinUSGraph /\ U e. V ) )
4	3	adantr 481	. . . . . . 7 |- ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) -> ( G e. FinUSGraph /\ U e. V ) )
5		simprl 794	. . . . . . 7 |- ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) -> M e. V )
6		simpr 477	. . . . . . . 8 |- ( ( M e. V /\ M =/= U ) -> M =/= U )
7	6	adantl 482	. . . . . . 7 |- ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) -> M =/= U )
8		df-nel 2898	. . . . . . . . . 10 |- ( M e/ ( G NeighbVtx U ) <-> -. M e. ( G NeighbVtx U ) )
9	8	biimpri 218	. . . . . . . . 9 |- ( -. M e. ( G NeighbVtx U ) -> M e/ ( G NeighbVtx U ) )
10	9	adantr 481	. . . . . . . 8 |- ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) -> M e/ ( G NeighbVtx U ) )
11	10	adantr 481	. . . . . . 7 |- ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) -> M e/ ( G NeighbVtx U ) )
12		hashnbusgrnn0.v	. . . . . . . 8 |- V = (Vtx ` G )
13	12	nbfusgrlevtxm2 26280	. . . . . . 7 |- ( ( ( G e. FinUSGraph /\ U e. V ) /\ ( M e. V /\ M =/= U /\ M e/ ( G NeighbVtx U ) ) ) -> ( # ` ( G NeighbVtx U ) ) <_ ( ( # ` V ) - 2 ) )
14	4, 5, 7, 11, 13	syl13anc 1328	. . . . . 6 |- ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) -> ( # ` ( G NeighbVtx U ) ) <_ ( ( # ` V ) - 2 ) )
15		breq1 4656	. . . . . . . . 9 |- ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> ( ( # ` ( G NeighbVtx U ) ) <_ ( ( # ` V ) - 2 ) <-> ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) ) )
16	15	adantl 482	. . . . . . . 8 |- ( ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) /\ ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) ) -> ( ( # ` ( G NeighbVtx U ) ) <_ ( ( # ` V ) - 2 ) <-> ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) ) )
17	12	fusgrvtxfi 26211	. . . . . . . . . . . 12 |- ( G e. FinUSGraph -> V e. Fin )
18		hashcl 13147	. . . . . . . . . . . 12 |- ( V e. Fin -> ( # ` V ) e. NN0 )
19		nn0re 11301	. . . . . . . . . . . . 13 |- ( ( # ` V ) e. NN0 -> ( # ` V ) e. RR )
20		1red 10055	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. RR -> 1 e. RR )
21		2re 11090	. . . . . . . . . . . . . . . 16 |- 2 e. RR
22	21	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. RR -> 2 e. RR )
23		id 22	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. RR -> ( # ` V ) e. RR )
24		1lt2 11194	. . . . . . . . . . . . . . . 16 |- 1 < 2
25	24	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. RR -> 1 < 2 )
26	20, 22, 23, 25	ltsub2dd 10640	. . . . . . . . . . . . . 14 |- ( ( # ` V ) e. RR -> ( ( # ` V ) - 2 ) < ( ( # ` V ) - 1 ) )
27	23, 22	resubcld 10458	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. RR -> ( ( # ` V ) - 2 ) e. RR )
28		peano2rem 10348	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. RR -> ( ( # ` V ) - 1 ) e. RR )
29	27, 28	ltnled 10184	. . . . . . . . . . . . . 14 |- ( ( # ` V ) e. RR -> ( ( ( # ` V ) - 2 ) < ( ( # ` V ) - 1 ) <-> -. ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) ) )
30	26, 29	mpbid 222	. . . . . . . . . . . . 13 |- ( ( # ` V ) e. RR -> -. ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) )
31	19, 30	syl 17	. . . . . . . . . . . 12 |- ( ( # ` V ) e. NN0 -> -. ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) )
32	17, 18, 31	3syl 18	. . . . . . . . . . 11 |- ( G e. FinUSGraph -> -. ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) )
33	32	pm2.21d 118	. . . . . . . . . 10 |- ( G e. FinUSGraph -> ( ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) -> M e. ( G NeighbVtx U ) ) )
34	33	adantr 481	. . . . . . . . 9 |- ( ( G e. FinUSGraph /\ U e. V ) -> ( ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) -> M e. ( G NeighbVtx U ) ) )
35	34	ad3antlr 767	. . . . . . . 8 |- ( ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) /\ ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) ) -> ( ( ( # ` V ) - 1 ) <_ ( ( # ` V ) - 2 ) -> M e. ( G NeighbVtx U ) ) )
36	16, 35	sylbid 230	. . . . . . 7 |- ( ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) /\ ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) ) -> ( ( # ` ( G NeighbVtx U ) ) <_ ( ( # ` V ) - 2 ) -> M e. ( G NeighbVtx U ) ) )
37	36	ex 450	. . . . . 6 |- ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> ( ( # ` ( G NeighbVtx U ) ) <_ ( ( # ` V ) - 2 ) -> M e. ( G NeighbVtx U ) ) ) )
38	14, 37	mpid 44	. . . . 5 |- ( ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) /\ ( M e. V /\ M =/= U ) ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> M e. ( G NeighbVtx U ) ) )
39	38	ex 450	. . . 4 |- ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) -> ( ( M e. V /\ M =/= U ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> M e. ( G NeighbVtx U ) ) ) )
40	39	com23 86	. . 3 |- ( ( -. M e. ( G NeighbVtx U ) /\ ( G e. FinUSGraph /\ U e. V ) ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= U ) -> M e. ( G NeighbVtx U ) ) ) )
41	40	ex 450	. 2 |- ( -. M e. ( G NeighbVtx U ) -> ( ( G e. FinUSGraph /\ U e. V ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= U ) -> M e. ( G NeighbVtx U ) ) ) ) )
42	2, 41	pm2.61i 176	1 |- ( ( G e. FinUSGraph /\ U e. V ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= U ) -> M e. ( G NeighbVtx U ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   1c1 9937   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   NN0cn0 11292   #chash 13117  Vtxcvtx 25874   FinUSGraph cfusgr 26208   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-fusgr 26209  df-nbgr 26228

This theorem is referenced by:  nbusgrvtxm1uvtx  26306