Theorem frgrwopreglem1 28577

Description: Lemma 1 for frgrwopreg 28588: 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 6770	. 2 ⊢ 𝑉 ∈ V
3		frgrwopreg.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
4		rabexg 5250	. . . 4 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ∈ V)
5	3, 4	eqeltrid 2843	. . 3 ⊢ (𝑉 ∈ V → 𝐴 ∈ V)
6		frgrwopreg.b	. . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
7		difexg 5246	. . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V)
8	6, 7	eqeltrid 2843	. . 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 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∖ cdif 3880  ‘cfv 6418  Vtxcvtx 27269  VtxDegcvtxdg 27735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6376  df-fv 6426

This theorem is referenced by:  frgrwopreg2  28584  frgrwopreglem5  28586  frgrwopreglem5ALT  28587  frgrwopreg  28588