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Theorem frgrwopreglem1 30332
Description: Lemma 1 for frgrwopreg 30343: 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 6919 . 2 𝑉 ∈ V
3 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
4 rabexg 5336 . . . 4 (𝑉 ∈ V → {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} ∈ V)
53, 4eqeltrid 2844 . . 3 (𝑉 ∈ V → 𝐴 ∈ V)
6 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
7 difexg 5328 . . . 4 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
86, 7eqeltrid 2844 . . 3 (𝑉 ∈ V → 𝐵 ∈ V)
95, 8jca 511 . 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 395   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  cdif 3947  cfv 6560  Vtxcvtx 29014  VtxDegcvtxdg 29484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513  df-fv 6568
This theorem is referenced by:  frgrwopreg2  30339  frgrwopreglem5  30341  frgrwopreglem5ALT  30342  frgrwopreg  30343
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