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Theorem 0vtxrgr 27357

Description: A null 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 487	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^) → 𝑘 ∈ ℕ₀^)
2		rzal 4452	. . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
3	2	ad2antlr 725	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
4		eqid 2821	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2821	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
6	4, 5	isrgr 27340	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ₀^) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ₀^ ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
7	6	adantlr 713	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ₀^ ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
8	1, 3, 7	mpbir2and 711	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺 RegGraph 𝑘)
9	8	ralrimiva 3182	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ₀^* 𝐺 RegGraph 𝑘)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∅c0 4290 class class class wbr 5065 ‘cfv 6354 ℕ₀^*cxnn0 11966 Vtxcvtx 26780 VtxDegcvtxdg 27246 RegGraph crgr 27336

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-iota 6313 df-fv 6362 df-rgr 27338

This theorem is referenced by: 0vtxrusgr 27358

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