Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rusisusgra | Structured version Visualization version GIF version |
Description: Any k-regular undirected simple graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) |
Ref | Expression |
---|---|
rusisusgra | ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → 𝑉 USGrph 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusgraprop 26456 | . 2 ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)) | |
2 | 1 | simp1d 1066 | 1 ⊢ (〈𝑉, 𝐸〉 RegUSGrph 𝐾 → 𝑉 USGrph 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℕ0cn0 11169 USGrph cusg 25859 VDeg cvdg 26420 RegUSGrph crusgra 26450 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-rgra 26451 df-rusgra 26452 |
This theorem is referenced by: rusgranumwwlkl1 26473 rusgranumwlkl1 26474 rusgranumwlkb1 26481 rusgra0edg 26482 rusgranumwlks 26483 rusgranumwlk 26484 rusgranumwwlkg 26486 numclwwlkovf2num 26612 numclwwlk1 26625 numclwwlkqhash 26627 numclwwlk3 26636 numclwwlk5 26639 numclwwlk6 26640 frgrareg 26644 |
Copyright terms: Public domain | W3C validator |