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| Mirrors > Home > MPE Home > Th. List > fusgrusgr | Structured version Visualization version GIF version | ||
| Description: A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
| Ref | Expression |
|---|---|
| fusgrusgr | ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | 1 | isfusgr 29335 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin)) |
| 3 | 2 | simplbi 497 | 1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6561 Fincfn 8985 Vtxcvtx 29013 USGraphcusgr 29166 FinUSGraphcfusgr 29333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-fusgr 29334 |
| This theorem is referenced by: fusgredgfi 29342 fusgrfisstep 29346 fusgrfupgrfs 29348 nbfiusgrfi 29392 vtxdgfusgrf 29515 usgruvtxvdb 29547 vdiscusgrb 29548 vdiscusgr 29549 fusgrn0eqdrusgr 29588 wlksnfi 29927 fusgrhashclwwlkn 30098 clwlksndivn 30105 fusgr2wsp2nb 30353 fusgreghash2wspv 30354 numclwwlk4 30405 clnbfiusgrfi 47830 |
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