Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 0vtxrusgr | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
0vtxrusgr | ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr0v 40467 | . . . 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 40776 | . . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑣 ∈ ℕ0* 𝐺 RegGraph 𝑣) | |
4 | breq2 4587 | . . . . . . . 8 ⊢ (𝑣 = 𝑘 → (𝐺 RegGraph 𝑣 ↔ 𝐺 RegGraph 𝑘)) | |
5 | 4 | rspccva 3281 | . . . . . . 7 ⊢ ((∀𝑣 ∈ ℕ0* 𝐺 RegGraph 𝑣 ∧ 𝑘 ∈ ℕ0*) → 𝐺 RegGraph 𝑘) |
6 | 5 | ex 449 | . . . . . 6 ⊢ (∀𝑣 ∈ ℕ0* 𝐺 RegGraph 𝑣 → (𝑘 ∈ ℕ0* → 𝐺 RegGraph 𝑘)) |
7 | 3, 6 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝑘 ∈ ℕ0* → 𝐺 RegGraph 𝑘)) |
8 | 7 | 3adant3 1074 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → (𝑘 ∈ ℕ0* → 𝐺 RegGraph 𝑘)) |
9 | 8 | imp 444 | . . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 RegGraph 𝑘) |
10 | isrusgr 40761 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ0*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘))) | |
11 | 10 | 3ad2antl1 1216 | . . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘))) |
12 | 2, 9, 11 | mpbir2and 959 | . 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 RegUSGraph 𝑘) |
13 | 12 | ralrimiva 2949 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 ℕ0*cxnn0 11240 Vtxcvtx 25673 iEdgciedg 25674 USGraph cusgr 40379 RegGraph crgr 40755 RegUSGraph crusgr 40756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-2 10956 df-uhgr 25724 df-upgr 25749 df-uspgr 40380 df-usgr 40381 df-rgr 40757 df-rusgr 40758 |
This theorem is referenced by: 0uhgrrusgr 40778 0grrusgr 40779 |
Copyright terms: Public domain | W3C validator |