Theorem nbhashuvtx1 25488

Description: If the number of the neighbors of a vertex in a finite 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.)

Assertion

Ref	Expression
nbhashuvtx1	|- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) )

Proof of Theorem nbhashuvtx1

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 |- ( M e. ( <. V , E >. Neighbors N ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) )
2	1	2a1d 27	. 2 |- ( M e. ( <. V , E >. Neighbors N ) -> ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) ) )
3		df-nel 2628	. . 3 |- ( M e/ ( <. V , E >. Neighbors N ) <-> -. M e. ( <. V , E >. Neighbors N ) )
4		nbgrassvwo2 25011	. . . . . . 7 |- ( ( V USGrph E /\ M e/ ( <. V , E >. Neighbors N ) ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) )
5	4	ex 435	. . . . . 6 |- ( V USGrph E -> ( M e/ ( <. V , E >. Neighbors N ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) )
6	5	3ad2ant1 1026	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( M e/ ( <. V , E >. Neighbors N ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) )
7		difexg 4573	. . . . . . 7 |- ( V e. Fin -> ( V \ { M , N } ) e. _V )
8	7	3ad2ant2 1027	. . . . . 6 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( V \ { M , N } ) e. _V )
9		hashss 12583	. . . . . . 7 |- ( ( ( V \ { M , N } ) e. _V /\ ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) )
10	9	ex 435	. . . . . 6 |- ( ( V \ { M , N } ) e. _V -> ( ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) -> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) ) )
11	8, 10	syl 17	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) -> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) ) )
12		simpl2 1009	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> V e. Fin )
13		simpl 458	. . . . . . . . . . . . 13 |- ( ( M e. V /\ M =/= N ) -> M e. V )
14		simp3 1007	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> N e. V )
15		prssi 4159	. . . . . . . . . . . . 13 |- ( ( M e. V /\ N e. V ) -> { M , N } C_ V )
16	13, 14, 15	syl2anr 480	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> { M , N } C_ V )
17		hashssdif 12584	. . . . . . . . . . . 12 |- ( ( V e. Fin /\ { M , N } C_ V ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - ( # ` { M , N } ) ) )
18	12, 16, 17	syl2anc 665	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - ( # ` { M , N } ) ) )
19		simprr 764	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> M =/= N )
20		hashprg 12569	. . . . . . . . . . . . . 14 |- ( ( M e. V /\ N e. V ) -> ( M =/= N <-> ( # ` { M , N } ) = 2 ) )
21	13, 14, 20	syl2anr 480	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( M =/= N <-> ( # ` { M , N } ) = 2 ) )
22	19, 21	mpbid 213	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` { M , N } ) = 2 )
23	22	oveq2d 6321	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` V ) - ( # ` { M , N } ) ) = ( ( # ` V ) - 2 ) )
24	18, 23	eqtrd 2470	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - 2 ) )
25	24	breq2d 4438	. . . . . . . . 9 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( # ` V ) - 2 ) ) )
26		nbhashnn0 25487	. . . . . . . . . . . . . 14 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. NN0 )
27	26	nn0zd 11038	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. ZZ )
28		hashcl 12535	. . . . . . . . . . . . . . 15 |- ( V e. Fin -> ( # ` V ) e. NN0 )
29		nn0z 10960	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. NN0 -> ( # ` V ) e. ZZ )
30		peano2zm 10980	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. ZZ -> ( ( # ` V ) - 1 ) e. ZZ )
31	28, 29, 30	3syl 18	. . . . . . . . . . . . . 14 |- ( V e. Fin -> ( ( # ` V ) - 1 ) e. ZZ )
32	31	3ad2ant2 1027	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` V ) - 1 ) e. ZZ )
33		zltlem1 10989	. . . . . . . . . . . . 13 |- ( ( ( # ` ( <. V , E >. Neighbors N ) ) e. ZZ /\ ( ( # ` V ) - 1 ) e. ZZ ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( ( # ` V ) - 1 ) - 1 ) ) )
34	27, 32, 33	syl2anc 665	. . . . . . . . . . . 12 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( ( # ` V ) - 1 ) - 1 ) ) )
35	28	nn0cnd 10927	. . . . . . . . . . . . . . . 16 |- ( V e. Fin -> ( # ` V ) e. CC )
36	35	3ad2ant2 1027	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` V ) e. CC )
37		1cnd 9658	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> 1 e. CC )
38	36, 37, 37	subsub4d 10016	. . . . . . . . . . . . . 14 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( ( # ` V ) - 1 ) - 1 ) = ( ( # ` V ) - ( 1 + 1 ) ) )
39		1p1e2 10723	. . . . . . . . . . . . . . 15 |- ( 1 + 1 ) = 2
40	39	oveq2i 6316	. . . . . . . . . . . . . 14 |- ( ( # ` V ) - ( 1 + 1 ) ) = ( ( # ` V ) - 2 )
41	38, 40	syl6eq 2486	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( ( # ` V ) - 1 ) - 1 ) = ( ( # ` V ) - 2 ) )
42	41	breq2d 4438	. . . . . . . . . . . 12 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( ( # ` V ) - 1 ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( # ` V ) - 2 ) ) )
43	34, 42	bitrd 256	. . . . . . . . . . 11 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( # ` V ) - 2 ) ) )
44	43	adantr 466	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( # ` V ) - 2 ) ) )
45	26	nn0red 10926	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. RR )
46	45	adantr 466	. . . . . . . . . . . . . 14 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. RR )
47	46	adantr 466	. . . . . . . . . . . . 13 |- ( ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) /\ ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. RR )
48		simpr 462	. . . . . . . . . . . . 13 |- ( ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) /\ ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) )
49	47, 48	ltned 9770	. . . . . . . . . . . 12 |- ( ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) /\ ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) =/= ( ( # ` V ) - 1 ) )
50	49	ex 435	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) -> ( # ` ( <. V , E >. Neighbors N ) ) =/= ( ( # ` V ) - 1 ) ) )
51		eqneqall 2638	. . . . . . . . . . . 12 |- ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) =/= ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) )
52	51	com12 32	. . . . . . . . . . 11 |- ( ( # ` ( <. V , E >. Neighbors N ) ) =/= ( ( # ` V ) - 1 ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) )
53	50, 52	syl6 34	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) ) )
54	44, 53	sylbird 238	. . . . . . . . 9 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( # ` V ) - 2 ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) ) )
55	25, 54	sylbid 218	. . . . . . . 8 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) ) )
56	55	ex 435	. . . . . . 7 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( M e. V /\ M =/= N ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) ) ) )
57	56	com23 81	. . . . . 6 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) -> ( ( M e. V /\ M =/= N ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) ) ) )
58	57	com34 86	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) ) )
59	6, 11, 58	3syld 57	. . . 4 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( M e/ ( <. V , E >. Neighbors N ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) ) )
60	59	com12 32	. . 3 |- ( M e/ ( <. V , E >. Neighbors N ) -> ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) ) )
61	3, 60	sylbir 216	. 2 |- ( -. M e. ( <. V , E >. Neighbors N ) -> ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) ) )
62	2, 61	pm2.61i 167	1 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   =/= wne 2625   e/ wnel 2626   _Vcvv 3087   \ cdif 3439   C_ wss 3442   {cpr 4004   <.cop 4008   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Fincfn 7577   CCcc 9536   RRcr 9537   1c1 9539   + caddc 9541   < clt 9674   <_ cle 9675   - cmin 9859   2c2 10659   NN0cn0 10869   ZZcz 10937   #chash 12512   USGrph cusg 24903   Neighbors cnbgra 24990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-xadd 11410  df-fz 11783  df-hash 12513  df-usgra 24906  df-nbgra 24993  df-vdgr 25467

This theorem is referenced by:  nbhashuvtx  25489