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| Mirrors > Home > MPE Home > Th. List > iscusgr | Structured version Visualization version GIF version | ||
| Description: The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.) |
| Ref | Expression |
|---|---|
| iscusgr | ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-cusgr 29429 | . 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph) | |
| 2 | 1 | elin2 4203 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 USGraphcusgr 29166 ComplGraphccplgr 29426 ComplUSGraphccusgr 29427 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-cusgr 29429 |
| This theorem is referenced by: cusgrusgr 29436 cusgrcplgr 29437 iscusgrvtx 29438 cusgruvtxb 29439 iscusgredg 29440 cusgr0 29443 cusgr0v 29445 cusgr1v 29448 cusgrop 29455 cusgrexi 29460 structtocusgr 29463 cusgrres 29466 |
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