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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 27121 | . 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph) | |
2 | 1 | elin2 4171 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∈ wcel 2105 USGraphcusgr 26861 ComplGraphccplgr 27118 ComplUSGraphccusgr 27119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-cusgr 27121 |
This theorem is referenced by: cusgrusgr 27128 cusgrcplgr 27129 iscusgrvtx 27130 cusgruvtxb 27131 iscusgredg 27132 cusgr0 27135 cusgr0v 27137 cusgr1v 27140 cusgrop 27147 cusgrexi 27152 structtocusgr 27155 cusgrres 27157 |
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