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Theorem frgrwopreglem1 28100
Description: Lemma 1 for frgrwopreg 28111: 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)
21fvexi 6663 . 2 𝑉 ∈ V
3 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
4 rabexg 5201 . . . 4 (𝑉 ∈ V → {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} ∈ V)
53, 4eqeltrid 2897 . . 3 (𝑉 ∈ V → 𝐴 ∈ V)
6 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
7 difexg 5198 . . . 4 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
86, 7eqeltrid 2897 . . 3 (𝑉 ∈ V → 𝐵 ∈ V)
95, 8jca 515 . 2 (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
102, 9ax-mp 5 1 (𝐴 ∈ V ∧ 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  cdif 3881  cfv 6328  Vtxcvtx 26792  VtxDegcvtxdg 27258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-uni 4804  df-iota 6287  df-fv 6336
This theorem is referenced by:  frgrwopreg2  28107  frgrwopreglem5  28109  frgrwopreglem5ALT  28110  frgrwopreg  28111
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