![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uhgredgn0 | Structured version Visualization version GIF version |
Description: An edge of a hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020.) |
Ref | Expression |
---|---|
uhgredgn0 | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edgval 26842 | . . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | eqid 2798 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2798 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
4 | 2, 3 | uhgrf 26855 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
5 | 4 | frnd 6494 | . . 3 ⊢ (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
6 | 1, 5 | eqsstrid 3963 | . 2 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 6 | sselda 3915 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∖ cdif 3878 ∅c0 4243 𝒫 cpw 4497 {csn 4525 dom cdm 5519 ran crn 5520 ‘cfv 6324 Vtxcvtx 26789 iEdgciedg 26790 Edgcedg 26840 UHGraphcuhgr 26849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-edg 26841 df-uhgr 26851 |
This theorem is referenced by: edguhgr 26922 uhgredgss 26924 uhgrvd00 27324 lfuhgr2 32478 loop1cycl 32497 |
Copyright terms: Public domain | W3C validator |