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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 | eleq1 2689 | . 2 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ ComplGraph ↔ 𝐺 ∈ ComplGraph)) | |
2 | df-cusgr 26232 | . 2 ⊢ ComplUSGraph = {𝑔 ∈ USGraph ∣ 𝑔 ∈ ComplGraph} | |
3 | 1, 2 | elrab2 3366 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∈ wcel 1990 USGraph cusgr 26044 ComplGraphccplgr 26226 ComplUSGraphccusgr 26227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-cusgr 26232 |
This theorem is referenced by: cusgrusgr 26315 cusgrcplgr 26316 iscusgrvtx 26317 cusgruvtxb 26318 iscusgredg 26319 cusgr0 26322 cusgr0v 26324 cusgr1v 26327 cusgrop 26334 cusgrexi 26339 structtocusgr 26342 cusgrres 26344 |
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