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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 27202 | . 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph) | |
2 | 1 | elin2 4124 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 USGraphcusgr 26942 ComplGraphccplgr 27199 ComplUSGraphccusgr 27200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-cusgr 27202 |
This theorem is referenced by: cusgrusgr 27209 cusgrcplgr 27210 iscusgrvtx 27211 cusgruvtxb 27212 iscusgredg 27213 cusgr0 27216 cusgr0v 27218 cusgr1v 27221 cusgrop 27228 cusgrexi 27233 structtocusgr 27236 cusgrres 27238 |
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