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Mirrors > Home > MPE Home > Th. List > rusgrusgr | Structured version Visualization version GIF version |
Description: A k-regular simple graph is a simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
Ref | Expression |
---|---|
rusgrusgr | ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusgrprop 27352 | . 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)) | |
2 | 1 | simpld 498 | 1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 class class class wbr 5030 USGraphcusgr 26942 RegGraph crgr 27345 RegUSGraph crusgr 27346 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rusgr 27348 |
This theorem is referenced by: finrusgrfusgr 27355 rusgr0edg 27759 rusgrnumwwlks 27760 rusgrnumwwlk 27761 rusgrnumwlkg 27763 numclwwlk1 28146 clwlknon2num 28153 numclwlk1lem1 28154 numclwlk1lem2 28155 |
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