0vtxrusgr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 0vtxrusgr		Structured version Visualization version GIF version

Theorem 0vtxrusgr 29664

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 29327	. . . 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 29663	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑣 ∈ ℕ₀^* 𝐺 RegGraph 𝑣)
4		breq2 5090	. . . . . . 7 ⊢ (𝑣 = 𝑘 → (𝐺 RegGraph 𝑣 ↔ 𝐺 RegGraph 𝑘))
5	4	rspccv 3562	. . . . . 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 1133	. . . 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 29648	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ₀^*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
10	9	3ad2antl1 1187	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
11	2, 8, 10	mpbir2and 714	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ₀^*) → 𝐺 RegUSGraph 𝑘)
12	11	ralrimiva 3130	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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∅c0 4274 class class class wbr 5086 ‘cfv 6493 ℕ₀^*cxnn0 12504 Vtxcvtx 29082 iEdgciedg 29083 USGraphcusgr 29235 RegGraph crgr 29642 RegUSGraph crusgr 29643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-i2m1 11100 ax-1ne0 11101 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-2 12238 df-uhgr 29144 df-upgr 29168 df-uspgr 29236 df-usgr 29237 df-rgr 29644 df-rusgr 29645

This theorem is referenced by: 0uhgrrusgr 29665 0grrusgr 29666

W3C validator