![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rusgrprop | Structured version Visualization version GIF version |
Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
Ref | Expression |
---|---|
rusgrprop | ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rusgr 27046 | . . 3 ⊢ RegUSGraph = {〈𝑔, 𝑘〉 ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)} | |
2 | 1 | bropaex12 5493 | . 2 ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V)) |
3 | isrusgr 27049 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))) | |
4 | 3 | biimpd 221 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))) |
5 | 2, 4 | mpcom 38 | 1 ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2050 Vcvv 3415 class class class wbr 4930 USGraphcusgr 26640 RegGraphcrgr 27043 RegUSGraphcrusgr 27044 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-br 4931 df-opab 4993 df-xp 5414 df-rusgr 27046 |
This theorem is referenced by: rusgrrgr 27051 rusgrusgr 27052 rusgrprop0 27055 |
Copyright terms: Public domain | W3C validator |