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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 27194 | . 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 USGraphcusgr 26928 ComplGraphccplgr 27185 ComplUSGraphccusgr 27186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-in 3943 df-cusgr 27188 |
This theorem is referenced by: cusgrres 27224 cusgrsizeindslem 27227 cusgrsizeinds 27228 cusgrsize 27230 cusgrrusgr 27357 cusgredgex 32363 cusgr3cyclex 32378 cusgracyclt3v 32398 |
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