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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 26210 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
3 | 2 | simprbi 480 | 1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 Fincfn 7955 Vtxcvtx 25874 USGraph cusgr 26044 FinUSGraph cfusgr 26208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-fusgr 26209 |
This theorem is referenced by: fusgrfupgrfs 26223 nbusgrvtxm1 26281 uvtxanm1nbgr 26305 cusgrm1rusgr 26478 wlksnfi 26802 fusgrhashclwwlkn 26956 clwwlksndivn 26957 fusgreghash2wsp 27202 numclwwlk3lem 27241 numclwwlk4 27244 |
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