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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 26760 | . 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph) | |
2 | 1 | elin2 4024 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2107 USGraphcusgr 26498 ComplGraphccplgr 26757 ComplUSGraphccusgr 26758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-v 3400 df-in 3799 df-cusgr 26760 |
This theorem is referenced by: cusgrusgr 26767 cusgrcplgr 26768 iscusgrvtx 26769 cusgruvtxb 26770 iscusgredg 26771 cusgr0 26774 cusgr0v 26776 cusgr1v 26779 cusgrop 26786 cusgrexi 26791 structtocusgr 26794 cusgrres 26796 |
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