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Mirrors > Home > MPE Home > Th. List > fusgrvtxfi | Structured version Visualization version GIF version |
Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.) |
Ref | Expression |
---|---|
isfusgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
fusgrvtxfi | ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfusgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | isfusgr 27027 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
3 | 2 | simprbi 497 | 1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 Fincfn 8497 Vtxcvtx 26708 USGraphcusgr 26861 FinUSGraphcfusgr 27025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-fusgr 27026 |
This theorem is referenced by: fusgrfupgrfs 27040 nbfusgrlevtxm1 27086 nbfusgrlevtxm2 27087 nbusgrvtxm1 27088 uvtxnm1nbgr 27113 cusgrm1rusgr 27291 wlksnfi 27613 fusgrhashclwwlkn 27785 clwwlkndivn 27786 fusgreghash2wsp 28044 numclwwlk3lem2 28090 numclwwlk4 28092 |
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