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Theorem frgrawopreglem1 26571
 Description: Lemma 1 for frgrawopreg 26576. 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 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrawopreglem1 (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem frgrawopreglem1
StepHypRef Expression
1 frisusgra 26519 . 2 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
2 usgrav 25867 . 2 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 frgrawopreg.a . . . . 5 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
4 rabexg 4739 . . . . 5 (𝑉 ∈ V → {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} ∈ V)
53, 4syl5eqel 2692 . . . 4 (𝑉 ∈ V → 𝐴 ∈ V)
6 frgrawopreg.b . . . . 5 𝐵 = (𝑉𝐴)
7 difexg 4735 . . . . 5 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
86, 7syl5eqel 2692 . . . 4 (𝑉 ∈ V → 𝐵 ∈ V)
95, 8jca 553 . . 3 (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
109adantr 480 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
111, 2, 103syl 18 1 (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   USGrph cusg 25859   VDeg cvdg 26420   FriendGrph cfrgra 26515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-usgra 25862  df-frgra 26516 This theorem is referenced by:  frgrawopreglem5  26575  frgrawopreg  26576  frgrawopreg1  26577  frgrawopreg2  26578
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