Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgraisrusgra Structured version   Unicode version

Theorem cusgraisrusgra 24761
 Description: A complete undirected simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
Assertion
Ref Expression
cusgraisrusgra ComplUSGrph RegUSGrph

Proof of Theorem cusgraisrusgra
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nbcusgra 24286 . . . 4 ComplUSGrph Neighbors
21ralrimiva 2881 . . 3 ComplUSGrph Neighbors
3 cusisusgra 24281 . . . . . . . 8 ComplUSGrph USGrph
43adantr 465 . . . . . . 7 ComplUSGrph USGrph
54adantr 465 . . . . . 6 ComplUSGrph Neighbors USGrph
6 hashnncl 12416 . . . . . . . . 9
7 nnm1nn0 10849 . . . . . . . . 9
86, 7syl6bir 229 . . . . . . . 8
98imp 429 . . . . . . 7
109ad2antlr 726 . . . . . 6 ComplUSGrph Neighbors
114anim1i 568 . . . . . . . . . . . 12 ComplUSGrph USGrph
1211adantr 465 . . . . . . . . . . 11 ComplUSGrph Neighbors USGrph
13 hashnbgravdg 24736 . . . . . . . . . . 11 USGrph Neighbors VDeg
1412, 13syl 16 . . . . . . . . . 10 ComplUSGrph Neighbors Neighbors VDeg
15 fveq2 5872 . . . . . . . . . . 11 Neighbors Neighbors
16 simprl 755 . . . . . . . . . . . 12 ComplUSGrph
17 hashdifsn 12457 . . . . . . . . . . . 12
1816, 17sylan 471 . . . . . . . . . . 11 ComplUSGrph
1915, 18sylan9eqr 2530 . . . . . . . . . 10 ComplUSGrph Neighbors Neighbors
2014, 19eqtr3d 2510 . . . . . . . . 9 ComplUSGrph Neighbors VDeg
2120ex 434 . . . . . . . 8 ComplUSGrph Neighbors VDeg
2221ralimdva 2875 . . . . . . 7 ComplUSGrph Neighbors VDeg
2322imp 429 . . . . . 6 ComplUSGrph Neighbors VDeg
24 usgrav 24161 . . . . . . . . . . 11 USGrph
253, 24syl 16 . . . . . . . . . 10 ComplUSGrph
2625adantr 465 . . . . . . . . 9 ComplUSGrph
27 ovex 6320 . . . . . . . . . 10
2827a1i 11 . . . . . . . . 9 ComplUSGrph
29 df-3an 975 . . . . . . . . 9
3026, 28, 29sylanbrc 664 . . . . . . . 8 ComplUSGrph
3130adantr 465 . . . . . . 7 ComplUSGrph Neighbors
32 isrusgra 24750 . . . . . . 7 RegUSGrph USGrph VDeg
3331, 32syl 16 . . . . . 6 ComplUSGrph Neighbors RegUSGrph USGrph VDeg
345, 10, 23, 33mpbir3and 1179 . . . . 5 ComplUSGrph Neighbors RegUSGrph
3534expcom 435 . . . 4 Neighbors ComplUSGrph RegUSGrph
3635expd 436 . . 3 Neighbors ComplUSGrph RegUSGrph
372, 36mpcom 36 . 2 ComplUSGrph RegUSGrph
38373impib 1194 1 ComplUSGrph RegUSGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1379   wcel 1767   wne 2662  wral 2817  cvv 3118   cdif 3478  c0 3790  csn 4033  cop 4039   class class class wbr 4453  cfv 5594  (class class class)co 6295  cfn 7528  c1 9505   cmin 9817  cn 10548  cn0 10807  chash 12385   USGrph cusg 24153   Neighbors cnbgra 24240   ComplUSGrph ccusgra 24241   VDeg cvdg 24716   RegUSGrph crusgra 24746 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-cusgra 24244  df-vdgr 24717  df-rgra 24747  df-rusgra 24748 This theorem is referenced by:  cusgraiffrusgra  24763
 Copyright terms: Public domain W3C validator