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Mirrors > Home > MPE Home > Th. List > 0vtxrgr | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
0vtxrgr | ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝑘 ∈ ℕ0*) | |
2 | rzal 4449 | . . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘) | |
3 | 2 | ad2antlr 723 | . . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘) |
4 | eqid 2818 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | eqid 2818 | . . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
6 | 4, 5 | isrgr 27268 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ0*) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘))) |
7 | 6 | adantlr 711 | . . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘))) |
8 | 1, 3, 7 | mpbir2and 709 | . 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 RegGraph 𝑘) |
9 | 8 | ralrimiva 3179 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 ℕ0*cxnn0 11955 Vtxcvtx 26708 VtxDegcvtxdg 27174 RegGraph crgr 27264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-iota 6307 df-fv 6356 df-rgr 27266 |
This theorem is referenced by: 0vtxrusgr 27286 |
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