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Theorem cusgraisrusgra 25511
 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 25036 . . . 4 ComplUSGrph Neighbors
21ralrimiva 2846 . . 3 ComplUSGrph Neighbors
3 cusisusgra 25031 . . . . . . . 8 ComplUSGrph USGrph
43adantr 466 . . . . . . 7 ComplUSGrph USGrph
54adantr 466 . . . . . 6 ComplUSGrph Neighbors USGrph
6 hashnncl 12544 . . . . . . . . 9
7 nnm1nn0 10911 . . . . . . . . 9
86, 7syl6bir 232 . . . . . . . 8
98imp 430 . . . . . . 7
109ad2antlr 731 . . . . . 6 ComplUSGrph Neighbors
114anim1i 570 . . . . . . . . . . . 12 ComplUSGrph USGrph
1211adantr 466 . . . . . . . . . . 11 ComplUSGrph Neighbors USGrph
13 hashnbgravdg 25486 . . . . . . . . . . 11 USGrph Neighbors VDeg
1412, 13syl 17 . . . . . . . . . 10 ComplUSGrph Neighbors Neighbors VDeg
15 fveq2 5881 . . . . . . . . . . 11 Neighbors Neighbors
16 simprl 762 . . . . . . . . . . . 12 ComplUSGrph
17 hashdifsn 12586 . . . . . . . . . . . 12
1816, 17sylan 473 . . . . . . . . . . 11 ComplUSGrph
1915, 18sylan9eqr 2492 . . . . . . . . . 10 ComplUSGrph Neighbors Neighbors
2014, 19eqtr3d 2472 . . . . . . . . 9 ComplUSGrph Neighbors VDeg
2120ex 435 . . . . . . . 8 ComplUSGrph Neighbors VDeg
2221ralimdva 2840 . . . . . . 7 ComplUSGrph Neighbors VDeg
2322imp 430 . . . . . 6 ComplUSGrph Neighbors VDeg
24 usgrav 24911 . . . . . . . . . . 11 USGrph
253, 24syl 17 . . . . . . . . . 10 ComplUSGrph
2625adantr 466 . . . . . . . . 9 ComplUSGrph
27 ovex 6333 . . . . . . . . . 10
2827a1i 11 . . . . . . . . 9 ComplUSGrph
29 df-3an 984 . . . . . . . . 9
3026, 28, 29sylanbrc 668 . . . . . . . 8 ComplUSGrph
3130adantr 466 . . . . . . 7 ComplUSGrph Neighbors
32 isrusgra 25500 . . . . . . 7 RegUSGrph USGrph VDeg
3331, 32syl 17 . . . . . 6 ComplUSGrph Neighbors RegUSGrph USGrph VDeg
345, 10, 23, 33mpbir3and 1188 . . . . 5 ComplUSGrph Neighbors RegUSGrph
3534expcom 436 . . . 4 Neighbors ComplUSGrph RegUSGrph
3635expd 437 . . 3 Neighbors ComplUSGrph RegUSGrph
372, 36mpcom 37 . 2 ComplUSGrph RegUSGrph
38373impib 1203 1 ComplUSGrph RegUSGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   w3a 982   wceq 1437   wcel 1870   wne 2625  wral 2782  cvv 3087   cdif 3439  c0 3767  csn 4002  cop 4008   class class class wbr 4426  cfv 5601  (class class class)co 6305  cfn 7577  c1 9539   cmin 9859  cn 10609  cn0 10869  chash 12512   USGrph cusg 24903   Neighbors cnbgra 24990   ComplUSGrph ccusgra 24991   VDeg cvdg 25466   RegUSGrph crusgra 25496 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-cusgra 24994  df-vdgr 25467  df-rgra 25497  df-rusgra 25498 This theorem is referenced by:  cusgraiffrusgra  25513
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