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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 2676 | . 2 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ ComplGraph ↔ 𝐺 ∈ ComplGraph)) | |
2 | df-cusgr 40558 | . 2 ⊢ ComplUSGraph = {𝑔 ∈ USGraph ∣ 𝑔 ∈ ComplGraph} | |
3 | 1, 2 | elrab2 3333 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 USGraph cusgr 40379 ComplGraphccplgr 40552 ComplUSGraphccusgr 40553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-cusgr 40558 |
This theorem is referenced by: cusgrusgr 40641 cusgrcplgr 40642 iscusgrvtx 40643 cusgruvtxb 40644 iscusgredg 40645 cusgr0 40648 cusgr0v 40650 cusgr1v 40653 cusgrop 40660 cusgrexi 40662 cusgrres 40664 |
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