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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 USGrph Neighbors Neighbors

Proof of Theorem nbhashuvtx1
StepHypRef Expression
1 ax-1 6 . . . 4 Neighbors Neighbors
21a1d 25 . . 3 Neighbors Neighbors Neighbors
32a1d 25 . 2 Neighbors USGrph Neighbors Neighbors
4 df-nel 2665 . . 3 Neighbors Neighbors
5 nbgrassvwo2 24261 . . . . . . 7 USGrph Neighbors Neighbors
65ex 434 . . . . . 6 USGrph Neighbors Neighbors
763ad2ant1 1017 . . . . 5 USGrph Neighbors Neighbors
8 difexg 4601 . . . . . . 7
983ad2ant2 1018 . . . . . 6 USGrph
10 hashss 12454 . . . . . . 7 Neighbors Neighbors
1110ex 434 . . . . . 6 Neighbors Neighbors
129, 11syl 16 . . . . 5 USGrph Neighbors Neighbors
13 simpl2 1000 . . . . . . . . . . . 12 USGrph
14 simpl 457 . . . . . . . . . . . . 13
15 simp3 998 . . . . . . . . . . . . 13 USGrph
16 prssi 4189 . . . . . . . . . . . . 13
1714, 15, 16syl2anr 478 . . . . . . . . . . . 12 USGrph
18 hashssdif 12455 . . . . . . . . . . . 12
1913, 17, 18syl2anc 661 . . . . . . . . . . 11 USGrph
20 simprr 756 . . . . . . . . . . . . 13 USGrph
21 hashprg 12440 . . . . . . . . . . . . . 14
2214, 15, 21syl2anr 478 . . . . . . . . . . . . 13 USGrph
2320, 22mpbid 210 . . . . . . . . . . . 12 USGrph
2423oveq2d 6311 . . . . . . . . . . 11 USGrph
2519, 24eqtrd 2508 . . . . . . . . . 10 USGrph
2625breq2d 4465 . . . . . . . . 9 USGrph Neighbors Neighbors
27 nbhashnn0 24737 . . . . . . . . . . . . . 14 USGrph Neighbors
28 nn0z 10899 . . . . . . . . . . . . . 14 Neighbors Neighbors
2927, 28syl 16 . . . . . . . . . . . . 13 USGrph Neighbors
30 hashcl 12408 . . . . . . . . . . . . . . 15
31 nn0z 10899 . . . . . . . . . . . . . . 15
32 peano2zm 10918 . . . . . . . . . . . . . . 15
3330, 31, 323syl 20 . . . . . . . . . . . . . 14
34333ad2ant2 1018 . . . . . . . . . . . . 13 USGrph
35 zltlem1 10927 . . . . . . . . . . . . 13 Neighbors Neighbors Neighbors
3629, 34, 35syl2anc 661 . . . . . . . . . . . 12 USGrph Neighbors Neighbors
3730nn0cnd 10866 . . . . . . . . . . . . . . . 16
38373ad2ant2 1018 . . . . . . . . . . . . . . 15 USGrph
39 ax-1cn 9562 . . . . . . . . . . . . . . . 16
4039a1i 11 . . . . . . . . . . . . . . 15 USGrph
4138, 40, 40subsub4d 9973 . . . . . . . . . . . . . 14 USGrph
42 1p1e2 10661 . . . . . . . . . . . . . . 15
4342oveq2i 6306 . . . . . . . . . . . . . 14
4441, 43syl6eq 2524 . . . . . . . . . . . . 13 USGrph
4544breq2d 4465 . . . . . . . . . . . 12 USGrph Neighbors Neighbors
4636, 45bitrd 253 . . . . . . . . . . 11 USGrph Neighbors Neighbors
4746adantr 465 . . . . . . . . . 10 USGrph Neighbors Neighbors
4827nn0red 10865 . . . . . . . . . . . . . . 15 USGrph Neighbors
4948adantr 465 . . . . . . . . . . . . . 14 USGrph Neighbors
5049adantr 465 . . . . . . . . . . . . 13 USGrph Neighbors Neighbors
51 simpr 461 . . . . . . . . . . . . 13 USGrph Neighbors Neighbors
5250, 51ltned 9732 . . . . . . . . . . . 12 USGrph Neighbors Neighbors
5352ex 434 . . . . . . . . . . 11 USGrph Neighbors Neighbors
54 eqneqall 2674 . . . . . . . . . . . 12 Neighbors Neighbors Neighbors
5554com12 31 . . . . . . . . . . 11 Neighbors Neighbors Neighbors
5653, 55syl6 33 . . . . . . . . . 10 USGrph Neighbors Neighbors Neighbors
5747, 56sylbird 235 . . . . . . . . 9 USGrph Neighbors Neighbors Neighbors
5826, 57sylbid 215 . . . . . . . 8 USGrph Neighbors Neighbors Neighbors
5958ex 434 . . . . . . 7 USGrph Neighbors Neighbors Neighbors
6059com23 78 . . . . . 6 USGrph Neighbors Neighbors Neighbors
6160com34 83 . . . . 5 USGrph Neighbors Neighbors Neighbors
627, 12, 613syld 55 . . . 4 USGrph Neighbors Neighbors Neighbors
6362com12 31 . . 3 Neighbors USGrph Neighbors Neighbors
644, 63sylbir 213 . 2 Neighbors USGrph Neighbors Neighbors
653, 64pm2.61i 164 1 USGrph Neighbors Neighbors
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   w3a 973   wceq 1379   wcel 1767   wne 2662   wnel 2663  cvv 3118   cdif 3478   wss 3481  cpr 4035  cop 4039   class class class wbr 4453  cfv 5594  (class class class)co 6295  cfn 7528  cc 9502  cr 9503  c1 9505   caddc 9507   clt 9640   cle 9641   cmin 9817  c2 10597  cn0 10807  cz 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
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