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Theorem 0vtxrusgr 29556

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

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

Distinct variable groups: 𝑘,𝐺 𝑘,𝑊

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

Step	Hyp	Ref	Expression
1		usgr0v 29219	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph)
2	1	adantr 480	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺 ∈ USGraph)
3		0vtxrgr 29555	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑣 ∈ ℕ₀^* 𝐺 RegGraph 𝑣)
4		breq2 5093	. . . . . . 7 ⊢ (𝑣 = 𝑘 → (𝐺 RegGraph 𝑣 ↔ 𝐺 RegGraph 𝑘))
5	4	rspccv 3569	. . . . . 6 ⊢ (∀𝑣 ∈ ℕ₀^* 𝐺 RegGraph 𝑣 → (𝑘 ∈ ℕ₀^* → 𝐺 RegGraph 𝑘))
6	3, 5	syl 17	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝑘 ∈ ℕ₀^* → 𝐺 RegGraph 𝑘))
7	6	3adant3 1132	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → (𝑘 ∈ ℕ₀^* → 𝐺 RegGraph 𝑘))
8	7	imp 406	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺 RegGraph 𝑘)
9		isrusgr 29540	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ₀^*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
10	9	3ad2antl1 1186	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
11	2, 8, 10	mpbir2and 713	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺 RegUSGraph 𝑘)
12	11	ralrimiva 3124	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ₀^* 𝐺 RegUSGraph 𝑘)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∅c0 4280 class class class wbr 5089 ‘cfv 6481 ℕ₀^*cxnn0 12454 Vtxcvtx 28974 iEdgciedg 28975 USGraphcusgr 29127 RegGraph crgr 29534 RegUSGraph crusgr 29535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-i2m1 11074 ax-1ne0 11075 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-2 12188 df-uhgr 29036 df-upgr 29060 df-uspgr 29128 df-usgr 29129 df-rgr 29536 df-rusgr 29537

This theorem is referenced by: 0uhgrrusgr 29557 0grrusgr 29558

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