![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 29302 | . 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph) | |
2 | 1 | elin2 4195 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2098 USGraphcusgr 29039 ComplGraphccplgr 29299 ComplUSGraphccusgr 29300 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-in 3951 df-cusgr 29302 |
This theorem is referenced by: cusgrusgr 29309 cusgrcplgr 29310 iscusgrvtx 29311 cusgruvtxb 29312 iscusgredg 29313 cusgr0 29316 cusgr0v 29318 cusgr1v 29321 cusgrop 29328 cusgrexi 29333 structtocusgr 29336 cusgrres 29339 |
Copyright terms: Public domain | W3C validator |