Theorem frgrwopreglem1 30793

Description: Lemma 1 for frgrwopreg 30804: 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 6896	. 2 ⊢ 𝑉 ∈ V
3		frgrwopreg.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
4		rabexg 5306	. . . 4 ⊢ (𝑉 ∈ V → {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ∈ V)
5	3, 4	eqeltrid 2866	. . 3 ⊢ (𝑉 ∈ V → 𝐴 ∈ V)
6		frgrwopreg.b	. . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
7		difexg 5298	. . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ 𝐴) ∈ V)
8	6, 7	eqeltrid 2866	. . 3 ⊢ (𝑉 ∈ V → 𝐵 ∈ V)
9	5, 8	jca 521	. 2 ⊢ (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	2, 9	ax-mp 5	1 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ∖ cdif 3899  ‘cfv 6537  Vtxcvtx 29454  VtxDegcvtxdg 29926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-pw 4562  df-sn 4588  df-pr 4590  df-uni 4871  df-iota 6493  df-fv 6545

This theorem is used by:  frgrwopreg2  30800  frgrwopreglem5  30802  frgrwopreglem5ALT  30803  frgrwopreg  30804