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Theorem frgrancvvdgeq 25169
 Description: In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y, are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Assertion
Ref Expression
frgrancvvdgeq FriendGrph Neighbors VDeg VDeg
Distinct variable groups:   ,,   ,,

Proof of Theorem frgrancvvdgeq
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeqlem9 25167 . 2 FriendGrph Neighbors Neighbors Neighbors
2 ovex 6324 . . . . . . . . . 10 Neighbors
3 ovex 6324 . . . . . . . . . 10 Neighbors
42, 3pm3.2i 455 . . . . . . . . 9 Neighbors Neighbors
5 hasheqf1oi 12426 . . . . . . . . 9 Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
64, 5mp1i 12 . . . . . . . 8 FriendGrph Neighbors Neighbors Neighbors Neighbors
76imim2d 52 . . . . . . 7 FriendGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
87imp31 432 . . . . . 6 FriendGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
9 frisusgra 25118 . . . . . . . . . 10 FriendGrph USGrph
109ad2antrr 725 . . . . . . . . 9 FriendGrph USGrph
11 simplr 755 . . . . . . . . 9 FriendGrph
1210, 11jca 532 . . . . . . . 8 FriendGrph USGrph
1312ad2antrr 725 . . . . . . 7 FriendGrph Neighbors Neighbors Neighbors Neighbors USGrph
14 hashnbgravdg 25039 . . . . . . 7 USGrph Neighbors VDeg
1513, 14syl 16 . . . . . 6 FriendGrph Neighbors Neighbors Neighbors Neighbors Neighbors VDeg
16 eldifi 3622 . . . . . . . . . 10
1716adantl 466 . . . . . . . . 9 FriendGrph
1810, 17jca 532 . . . . . . . 8 FriendGrph USGrph
1918ad2antrr 725 . . . . . . 7 FriendGrph Neighbors Neighbors Neighbors Neighbors USGrph
20 hashnbgravdg 25039 . . . . . . 7 USGrph Neighbors VDeg
2119, 20syl 16 . . . . . 6 FriendGrph Neighbors Neighbors Neighbors Neighbors Neighbors VDeg
228, 15, 213eqtr3d 2506 . . . . 5 FriendGrph Neighbors Neighbors Neighbors Neighbors VDeg VDeg
2322exp31 604 . . . 4 FriendGrph Neighbors Neighbors Neighbors Neighbors VDeg VDeg
2423ralimdva 2865 . . 3 FriendGrph Neighbors Neighbors Neighbors Neighbors VDeg VDeg
2524ralimdva 2865 . 2 FriendGrph Neighbors Neighbors Neighbors Neighbors VDeg VDeg
261, 25mpd 15 1 FriendGrph Neighbors VDeg VDeg
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395  wex 1613   wcel 1819   wnel 2653  wral 2807  cvv 3109   cdif 3468  csn 4032  cop 4038   class class class wbr 4456  wf1o 5593  cfv 5594  (class class class)co 6296  chash 12407   USGrph cusg 24456   Neighbors cnbgra 24543   VDeg cvdg 25019   FriendGrph cfrgra 25114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-xadd 11344  df-fz 11698  df-hash 12408  df-usgra 24459  df-nbgra 24546  df-vdgr 25020  df-frgra 25115 This theorem is referenced by:  frgrawopreglem4  25173
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