Theorem nbhashuvtx1 25692

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 2636	. . 3 |- ( M e/ ( <. V , E >. Neighbors N ) <-> -. M e. ( <. V , E >. Neighbors N ) )
4		nbgrassvwo2 25215	. . . . . . 7 |- ( ( V USGrph E /\ M e/ ( <. V , E >. Neighbors N ) ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) )
5	4	ex 440	. . . . . 6 |- ( V USGrph E -> ( M e/ ( <. V , E >. Neighbors N ) -> ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) )
6	5	3ad2ant1 1035	. . . . 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 4565	. . . . . . 7 |- ( V e. Fin -> ( V \ { M , N } ) e. _V )
8	7	3ad2ant2 1036	. . . . . 6 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( V \ { M , N } ) e. _V )
9		hashss 12618	. . . . . . 7 |- ( ( ( V \ { M , N } ) e. _V /\ ( <. V , E >. Neighbors N ) C_ ( V \ { M , N } ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) <_ ( # ` ( V \ { M , N } ) ) )
10	9	ex 440	. . . . . 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 1018	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> V e. Fin )
13		simpl 463	. . . . . . . . . . . . 13 |- ( ( M e. V /\ M =/= N ) -> M e. V )
14		simp3 1016	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> N e. V )
15		prssi 4141	. . . . . . . . . . . . 13 |- ( ( M e. V /\ N e. V ) -> { M , N } C_ V )
16	13, 14, 15	syl2anr 485	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> { M , N } C_ V )
17		hashssdif 12621	. . . . . . . . . . . 12 |- ( ( V e. Fin /\ { M , N } C_ V ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - ( # ` { M , N } ) ) )
18	12, 16, 17	syl2anc 671	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - ( # ` { M , N } ) ) )
19		simprr 771	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> M =/= N )
20		hashprg 12604	. . . . . . . . . . . . . 14 |- ( ( M e. V /\ N e. V ) -> ( M =/= N <-> ( # ` { M , N } ) = 2 ) )
21	13, 14, 20	syl2anr 485	. . . . . . . . . . . . 13 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( M =/= N <-> ( # ` { M , N } ) = 2 ) )
22	19, 21	mpbid 215	. . . . . . . . . . . 12 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` { M , N } ) = 2 )
23	22	oveq2d 6331	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( ( # ` V ) - ( # ` { M , N } ) ) = ( ( # ` V ) - 2 ) )
24	18, 23	eqtrd 2496	. . . . . . . . . 10 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( V \ { M , N } ) ) = ( ( # ` V ) - 2 ) )
25	24	breq2d 4428	. . . . . . . . 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 25691	. . . . . . . . . . . . . 14 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. NN0 )
27	26	nn0zd 11067	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. ZZ )
28		hashcl 12570	. . . . . . . . . . . . . . 15 |- ( V e. Fin -> ( # ` V ) e. NN0 )
29		nn0z 10989	. . . . . . . . . . . . . . 15 |- ( ( # ` V ) e. NN0 -> ( # ` V ) e. ZZ )
30		peano2zm 11009	. . . . . . . . . . . . . . 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 1036	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( # ` V ) - 1 ) e. ZZ )
33		zltlem1 11018	. . . . . . . . . . . . 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 671	. . . . . . . . . . . 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 10956	. . . . . . . . . . . . . . . 16 |- ( V e. Fin -> ( # ` V ) e. CC )
36	35	3ad2ant2 1036	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` V ) e. CC )
37		1cnd 9685	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> 1 e. CC )
38	36, 37, 37	subsub4d 10043	. . . . . . . . . . . . . 14 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( ( # ` V ) - 1 ) - 1 ) = ( ( # ` V ) - ( 1 + 1 ) ) )
39		1p1e2 10751	. . . . . . . . . . . . . . 15 |- ( 1 + 1 ) = 2
40	39	oveq2i 6326	. . . . . . . . . . . . . 14 |- ( ( # ` V ) - ( 1 + 1 ) ) = ( ( # ` V ) - 2 )
41	38, 40	syl6eq 2512	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( ( ( # ` V ) - 1 ) - 1 ) = ( ( # ` V ) - 2 ) )
42	41	breq2d 4428	. . . . . . . . . . . 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 261	. . . . . . . . . . 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 471	. . . . . . . . . 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 10955	. . . . . . . . . . . . . . 15 |- ( ( V USGrph E /\ V e. Fin /\ N e. V ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. RR )
46	45	adantr 471	. . . . . . . . . . . . . 14 |- ( ( ( V USGrph E /\ V e. Fin /\ N e. V ) /\ ( M e. V /\ M =/= N ) ) -> ( # ` ( <. V , E >. Neighbors N ) ) e. RR )
47	46	adantr 471	. . . . . . . . . . . . 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 467	. . . . . . . . . . . . 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 9797	. . . . . . . . . . . 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 440	. . . . . . . . . . 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 2646	. . . . . . . . . . . 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 243	. . . . . . . . 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 223	. . . . . . . 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 440	. . . . . . 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 218	. 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 169	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 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   e/ wnel 2634   _Vcvv 3057   \ cdif 3413   C_ wss 3416   {cpr 3982   <.cop 3986   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Fincfn 7595   CCcc 9563   RRcr 9564   1c1 9566   + caddc 9568   < clt 9701   <_ cle 9702   - cmin 9886   2c2 10687   NN0cn0 10898   ZZcz 10966   #chash 12547   USGrph cusg 25106   Neighbors cnbgra 25194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-xadd 11439  df-fz 11814  df-hash 12548  df-usgra 25109  df-nbgra 25197  df-vdgr 25671

This theorem is referenced by:  nbhashuvtx  25693