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Theorem 0vtxrgr 26875

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^* 𝐺RegGraph𝑘)

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 479	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^) → 𝑘 ∈ ℕ₀^)
2		rzal 4296	. . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
3	2	ad2antlr 720	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
4		eqid 2826	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2826	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
6	4, 5	isrgr 26858	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ₀^) → (𝐺RegGraph𝑘 ↔ (𝑘 ∈ ℕ₀^ ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
7	6	adantlr 708	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^) → (𝐺RegGraph𝑘 ↔ (𝑘 ∈ ℕ₀^ ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
8	1, 3, 7	mpbir2and 706	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺RegGraph𝑘)
9	8	ralrimiva 3176	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ₀^* 𝐺RegGraph𝑘)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ∅c0 4145 class class class wbr 4874 ‘cfv 6124 ℕ₀^*cxnn0 11691 Vtxcvtx 26295 VtxDegcvtxdg 26764 RegGraphcrgr 26854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-iota 6087 df-fv 6132 df-rgr 26856

This theorem is referenced by: 0vtxrusgr 26876

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