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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 27202 | . 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 2114 USGraphcusgr 26936 ComplGraphccplgr 27193 ComplUSGraphccusgr 27194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-cusgr 27196 |
This theorem is referenced by: cusgrres 27232 cusgrsizeindslem 27235 cusgrsizeinds 27236 cusgrsize 27238 cusgrrusgr 27365 cusgredgex 32370 cusgr3cyclex 32385 cusgracyclt3v 32405 |
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