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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 27347 | . 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)) | |
2 | 1 | simpld 497 | 1 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 class class class wbr 5069 USGraphcusgr 26937 RegGraph crgr 27340 RegUSGraph crusgr 27341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rusgr 27343 |
This theorem is referenced by: finrusgrfusgr 27350 rusgr0edg 27755 rusgrnumwwlks 27756 rusgrnumwwlk 27757 rusgrnumwlkg 27759 numclwwlk1 28143 clwlknon2num 28150 numclwlk1lem1 28151 numclwlk1lem2 28152 |
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