Theorem frgrwopreglem1 30298

Description: Lemma 1 for frgrwopreg 30309: 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 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
frgrwopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)

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

Step	Hyp	Ref	Expression
1		frgrwopreg.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6895	. 2 ⊢ 𝑉 ∈ V
3		frgrwopreg.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
4		rabexg 5312	. . . 4 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ∈ V)
5	3, 4	eqeltrid 2839	. . 3 ⊢ (𝑉 ∈ V → 𝐴 ∈ V)
6		frgrwopreg.b	. . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
7		difexg 5304	. . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V)
8	6, 7	eqeltrid 2839	. . 3 ⊢ (𝑉 ∈ V → 𝐵 ∈ V)
9	5, 8	jca 511	. 2 ⊢ (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	2, 9	ax-mp 5	1 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464   ∖ cdif 3928  ‘cfv 6536  Vtxcvtx 28980  VtxDegcvtxdg 29450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-pw 4582  df-sn 4607  df-pr 4609  df-uni 4889  df-iota 6489  df-fv 6544

This theorem is referenced by:  frgrwopreg2  30305  frgrwopreglem5  30307  frgrwopreglem5ALT  30308  frgrwopreg  30309