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Mirrors > Home > MPE Home > Th. List > cusgrusgr | Structured version Visualization version GIF version |
Description: A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
Ref | Expression |
---|---|
cusgrusgr | ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscusgr 29183 | . 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 USGraphcusgr 28917 ComplGraphccplgr 29174 ComplUSGraphccusgr 29175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-cusgr 29177 |
This theorem is referenced by: cusgrres 29214 cusgrsizeindslem 29217 cusgrsizeinds 29218 cusgrsize 29220 cusgrrusgr 29347 cusgredgex 34640 cusgr3cyclex 34655 cusgracyclt3v 34675 |
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