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Theorem 0vtxrusgr 27052

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^* 𝐺RegUSGraph𝑘)

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 26716	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph)
2	1	adantr 473	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺 ∈ USGraph)
3		0vtxrgr 27051	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑣 ∈ ℕ₀^* 𝐺RegGraph𝑣)
4		breq2 4927	. . . . . . 7 ⊢ (𝑣 = 𝑘 → (𝐺RegGraph𝑣 ↔ 𝐺RegGraph𝑘))
5	4	rspccv 3526	. . . . . 6 ⊢ (∀𝑣 ∈ ℕ₀^* 𝐺RegGraph𝑣 → (𝑘 ∈ ℕ₀^* → 𝐺RegGraph𝑘))
6	3, 5	syl 17	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝑘 ∈ ℕ₀^* → 𝐺RegGraph𝑘))
7	6	3adant3 1112	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → (𝑘 ∈ ℕ₀^* → 𝐺RegGraph𝑘))
8	7	imp 398	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺RegGraph𝑘)
9		isrusgr 27036	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ₀^*) → (𝐺RegUSGraph𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝑘)))
10	9	3ad2antl1 1165	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → (𝐺RegUSGraph𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝑘)))
11	2, 8, 10	mpbir2and 700	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺RegUSGraph𝑘)
12	11	ralrimiva 3126	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ₀^* 𝐺RegUSGraph𝑘)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∅c0 4173 class class class wbr 4923 ‘cfv 6182 ℕ₀^*cxnn0 11772 Vtxcvtx 26474 iEdgciedg 26475 USGraphcusgr 26627 RegGraphcrgr 27030 RegUSGraphcrusgr 27031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-i2m1 10395 ax-1ne0 10396 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-2 11496 df-uhgr 26536 df-upgr 26560 df-uspgr 26628 df-usgr 26629 df-rgr 27032 df-rusgr 27033

This theorem is referenced by: 0uhgrrusgr 27053 0grrusgr 27054

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