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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 2736 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | isfusgr 27360 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin)) |
3 | 2 | simplbi 501 | 1 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ‘cfv 6358 Fincfn 8604 Vtxcvtx 27041 USGraphcusgr 27194 FinUSGraphcfusgr 27358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-fusgr 27359 |
This theorem is referenced by: fusgredgfi 27367 fusgrfisstep 27371 fusgrfupgrfs 27373 nbfiusgrfi 27417 vtxdgfusgrf 27539 usgruvtxvdb 27571 vdiscusgrb 27572 vdiscusgr 27573 fusgrn0eqdrusgr 27612 wlksnfi 27945 fusgrhashclwwlkn 28116 clwlksndivn 28123 fusgr2wsp2nb 28371 fusgreghash2wspv 28372 numclwwlk4 28423 |
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