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Theorem frgrwopreglem1 27167
Description: Lemma 1 for frgrwopreg 27172: 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 6199 . . 3 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2696 . 2 𝑉 ∈ V
4 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
5 rabexg 4810 . . . 4 (𝑉 ∈ V → {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} ∈ V)
64, 5syl5eqel 2704 . . 3 (𝑉 ∈ V → 𝐴 ∈ V)
7 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
8 difexg 4806 . . . 4 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
97, 8syl5eqel 2704 . . 3 (𝑉 ∈ V → 𝐵 ∈ V)
106, 9jca 554 . 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 384   = wceq 1482  wcel 1989  {crab 2915  Vcvv 3198  cdif 3569  cfv 5886  Vtxcvtx 25868  VtxDegcvtxdg 26355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-sn 4176  df-pr 4178  df-uni 4435  df-iota 5849  df-fv 5894
This theorem is referenced by:  frgrwopreglem5  27171  frgrwopreg  27172  frgrwopreg2  27174
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