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| Mirrors > Home > MPE Home > Th. List > cusgrcplgr | Structured version Visualization version GIF version | ||
| Description: A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.) |
| Ref | Expression |
|---|---|
| cusgrcplgr | ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscusgr 29491 | . 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
| 2 | 1 | simprbi 496 | 1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2113 USGraphcusgr 29222 ComplGraphccplgr 29482 ComplUSGraphccusgr 29483 |
| 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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1544 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-v 3442 df-in 3908 df-cusgr 29485 |
| This theorem is referenced by: cusgrsizeindslem 29525 cusgrrusgr 29655 cusgredgex 35316 |
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