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Theorem frgrawopreglem1 25617
 Description: Lemma 1 for frgrawopreg 25622. In a friendship graph, the classes A and B are sets. The definition of A and B corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a VDeg
frgrawopreg.b
Assertion
Ref Expression
frgrawopreglem1 FriendGrph
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem frgrawopreglem1
StepHypRef Expression
1 frisusgra 25565 . 2 FriendGrph USGrph
2 usgrav 24911 . 2 USGrph
3 frgrawopreg.a . . . . 5 VDeg
4 rabexg 4575 . . . . 5 VDeg
53, 4syl5eqel 2521 . . . 4
6 frgrawopreg.b . . . . 5
7 difexg 4573 . . . . 5
86, 7syl5eqel 2521 . . . 4
95, 8jca 534 . . 3
109adantr 466 . 2
111, 2, 103syl 18 1 FriendGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1870  crab 2786  cvv 3087   cdif 3439   class class class wbr 4426  cfv 5601  (class class class)co 6305   USGrph cusg 24903   VDeg cvdg 25466   FriendGrph cfrgra 25561 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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  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-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-dm 4864  df-rn 4865  df-usgra 24906  df-frgra 25562 This theorem is referenced by:  frgrawopreglem5  25621  frgrawopreg  25622  frgrawopreg1  25623  frgrawopreg2  25624
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