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Theorem frgrwopreglem1 41481
 Description: Lemma 1 for frgrwopreg 41486: 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.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrwopreglem1 (𝐴 ∈ V ∧ 𝐵 ∈ V)
Distinct variable group:   𝑥,𝑉
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐷(𝑥)   𝐺(𝑥)   𝐾(𝑥)

Proof of Theorem frgrwopreglem1
StepHypRef Expression
1 frgrwopreg.v . . 3 𝑉 = (Vtx‘𝐺)
2 fvex 6113 . . 3 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2684 . 2 𝑉 ∈ V
4 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
5 rabexg 4739 . . . 4 (𝑉 ∈ V → {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} ∈ V)
64, 5syl5eqel 2692 . . 3 (𝑉 ∈ V → 𝐴 ∈ V)
7 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
8 difexg 4735 . . . 4 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
97, 8syl5eqel 2692 . . 3 (𝑉 ∈ V → 𝐵 ∈ V)
106, 9jca 553 . 2 (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
113, 10ax-mp 5 1 (𝐴 ∈ V ∧ 𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537  ‘cfv 5804  Vtxcvtx 25673  VtxDegcvtxdg 40681 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-fv 5812 This theorem is referenced by:  frgrwopreglem5  41485  frgrwopreg  41486  frgrwopreg2  41488
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