0vtxrgr - Mathbox for Alexander van der Vekens

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Theorem 0vtxrgr 40776

Description: A graph with no vertices is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)

**Assertion**
Ref	Expression
0vtxrgr	⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ₀^* 𝐺 RegGraph 𝑘)

Distinct variable groups: 𝑘,𝐺 𝑘,𝑊

Proof of Theorem 0vtxrgr Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 476	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^) → 𝑘 ∈ ℕ₀^)
2		rzal 4025	. . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
3	2	ad2antlr 759	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
4		eqid 2610	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2610	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
6	4, 5	isrgr 40759	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ₀^) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ₀^ ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
7	6	adantlr 747	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ₀^ ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
8	1, 3, 7	mpbir2and 959	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺 RegGraph 𝑘)
9	8	ralrimiva 2949	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ₀^* 𝐺 RegGraph 𝑘)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 ℕ₀^*cxnn0 11240 Vtxcvtx 25673 VtxDegcvtxdg 40681 RegGraph crgr 40755

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-iota 5768 df-fv 5812 df-rgr 40757

This theorem is referenced by: 0vtxrusgr 40777

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