Theorem nbhashuvtx1 24738

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	. . . 4 |- ( M e. ( <. V , E >. Neighbors N ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) )
2	1	a1d 25	. . 3 |- ( M e. ( <. V , E >. Neighbors N ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( M e. V /\ M =/= N ) -> M e. ( <. V , E >. Neighbors N ) ) ) )
3	2	a1d 25	. 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 ) ) ) ) )
4		df-nel 2665	. . 3 |- ( M e/ ( <. V , E >. Neighbors N ) <-> -. M e. ( <. V , E >. Neighbors N ) )
5		nbgrassvwo2 24261	. . . . . . 7 |- ( ( V USGrph E /\ M e/ ( <. V , E >. Neighbors N ) ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) )
6	5	ex 434	. . . . . 6 |- ( V USGrph E -> ( M e/ ( <. V , E >. Neighbors N ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) )
7	6	3ad2ant1 1017	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( M e/ ( <. V , E >. Neighbors N ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) )
8		difexg 4601	. . . . . . 7 |- ( V e. Fin -> ( V \ { M , N } ) e. _V )
9	8	3ad2ant2 1018	. . . . . 6 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( V \ { M , N } ) e. _V )
10		hashss 12454	. . . . . . 7 |- ( ( ( V \ { M , N } ) e. _V /\ ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) )
11	10	ex 434	. . . . . 6 |- ( ( V \ { M , N } ) e. _V -> ( ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) -> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) ) )
12	9, 11	syl 16	. . . . 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 } ) ) ) )
13		simpl2 1000	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> V e. Fin )
14		simpl 457	. . . . . . . . . . . . 13 |- ( ( M e. V /\ M =/= N ) -> M e. V )
15		simp3 998	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> N e. V )
16		prssi 4189	. . . . . . . . . . . . 13 |- ( ( M e. V /\ N e. V ) -> { M , N } C_ V )
17	14, 15, 16	syl2anr 478	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> { M , N } C_ V )
18		hashssdif 12455	. . . . . . . . . . . 12 |- ( ( V e. Fin /\ { M , N } C_ V ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - ( # ` { M , N } ) ) )
19	13, 17, 18	syl2anc 661	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - ( # ` { M , N } ) ) )
20		simprr 756	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> M =/= N )
21		hashprg 12440	. . . . . . . . . . . . . 14 |- ( ( M e. V /\ N e. V ) -> ( M =/= N <-> ( # ` { M , N } ) = 2 ) )
22	14, 15, 21	syl2anr 478	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( M =/= N <-> ( # ` { M , N } ) = 2 ) )
23	20, 22	mpbid 210	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` { M , N } ) = 2 )
24	23	oveq2d 6311	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` V ) - ( # ` { M , N } ) ) = ( ( # ` V ) - 2 ) )
25	19, 24	eqtrd 2508	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - 2 ) )
26	25	breq2d 4465	. . . . . . . . 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 ) ) )
27		nbhashnn0 24737	. . . . . . . . . . . . . 14 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. NN0 )
28		nn0z 10899	. . . . . . . . . . . . . 14 |- ( ( # ` ( <. V , E >. Neighbors N ) ) e. NN0 -> ( # ` ( <. V , E >. Neighbors N ) ) e. ZZ )
29	27, 28	syl 16	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. ZZ )
30		hashcl 12408	. . . . . . . . . . . . . . 15 |- ( V e. Fin -> ( # ` V ) e. NN0 )
31		nn0z 10899	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. NN0 -> ( # ` V ) e. ZZ )
32		peano2zm 10918	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. ZZ -> ( ( # ` V ) - 1 ) e. ZZ )
33	30, 31, 32	3syl 20	. . . . . . . . . . . . . 14 |- ( V e. Fin -> ( ( # ` V ) - 1 ) e. ZZ )
34	33	3ad2ant2 1018	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` V ) - 1 ) e. ZZ )
35		zltlem1 10927	. . . . . . . . . . . . 13 |- ( ( ( # ` ( <. V , E >. Neighbors N ) ) e. ZZ /\ ( ( # ` V ) - 1 ) e. ZZ ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( ( # ` V ) - 1 ) - 1 ) ) )
36	29, 34, 35	syl2anc 661	. . . . . . . . . . . 12 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( ( # ` V ) - 1 ) - 1 ) ) )
37	30	nn0cnd 10866	. . . . . . . . . . . . . . . 16 |- ( V e. Fin -> ( # ` V ) e. CC )
38	37	3ad2ant2 1018	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` V ) e. CC )
39		ax-1cn 9562	. . . . . . . . . . . . . . . 16 |- 1 e. CC
40	39	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> 1 e. CC )
41	38, 40, 40	subsub4d 9973	. . . . . . . . . . . . . 14 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( ( # ` V ) - 1 ) - 1 ) = ( ( # ` V ) - ( 1 + 1 ) ) )
42		1p1e2 10661	. . . . . . . . . . . . . . 15 |- ( 1 + 1 ) = 2
43	42	oveq2i 6306	. . . . . . . . . . . . . 14 |- ( ( # ` V ) - ( 1 + 1 ) ) = ( ( # ` V ) - 2 )
44	41, 43	syl6eq 2524	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( ( # ` V ) - 1 ) - 1 ) = ( ( # ` V ) - 2 ) )
45	44	breq2d 4465	. . . . . . . . . . . 12 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( ( # ` V ) - 1 ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( # ` V ) - 2 ) ) )
46	36, 45	bitrd 253	. . . . . . . . . . 11 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) < ( ( # ` V ) - 1 ) <-> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( ( # ` V ) - 2 ) ) )
47	46	adantr 465	. . . . . . . . . 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 ) ) )
48	27	nn0red 10865	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. RR )
49	48	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. RR )
50	49	adantr 465	. . . . . . . . . . . . 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 )
51		simpr 461	. . . . . . . . . . . . 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 ) )
52	50, 51	ltned 9732	. . . . . . . . . . . 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 ) )
53	52	ex 434	. . . . . . . . . . 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 ) ) )
54		eqneqall 2674	. . . . . . . . . . . 12 |- ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) =/= ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) )
55	54	com12 31	. . . . . . . . . . 11 |- ( ( # ` ( <. V , E >. Neighbors N ) ) =/= ( ( # ` V ) - 1 ) -> ( ( # ` ( <. V , E >. Neighbors N ) ) = ( ( # ` V ) - 1 ) -> M e. ( <. V , E >. Neighbors N ) ) )
56	53, 55	syl6 33	. . . . . . . . . 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 ) ) ) )
57	47, 56	sylbird 235	. . . . . . . . 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 ) ) ) )
58	26, 57	sylbid 215	. . . . . . . 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 ) ) ) )
59	58	ex 434	. . . . . . 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 ) ) ) ) )
60	59	com23 78	. . . . . 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 ) ) ) ) )
61	60	com34 83	. . . . 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 ) ) ) ) )
62	7, 12, 61	3syld 55	. . . 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 ) ) ) ) )
63	62	com12 31	. . 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 ) ) ) ) )
64	4, 63	sylbir 213	. 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 ) ) ) ) )
65	3, 64	pm2.61i 164	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 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   e/ wnel 2663   _Vcvv 3118   \ cdif 3478   C_ wss 3481   {cpr 4035   <.cop 4039   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   RRcr 9503   1c1 9505   + caddc 9507   < clt 9640   <_ cle 9641   - cmin 9817   2c2 10597   NN0cn0 10807   ZZcz 10876   #chash 12385   USGrph cusg 24153   Neighbors cnbgra 24240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-xadd 11331  df-fz 11685  df-hash 12386  df-usgra 24156  df-nbgra 24243  df-vdgr 24717

This theorem is referenced by:  nbhashuvtx  24739