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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 26826 | . . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | eqid 2819 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2819 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
4 | 2, 3 | uhgrf 26839 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
5 | 4 | frnd 6514 | . . 3 ⊢ (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
6 | 1, 5 | eqsstrid 4013 | . 2 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 6 | sselda 3965 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2108 ∖ cdif 3931 ∅c0 4289 𝒫 cpw 4537 {csn 4559 dom cdm 5548 ran crn 5549 ‘cfv 6348 Vtxcvtx 26773 iEdgciedg 26774 Edgcedg 26824 UHGraphcuhgr 26833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-edg 26825 df-uhgr 26835 |
This theorem is referenced by: edguhgr 26906 uhgredgss 26908 uhgrvd00 27308 lfuhgr2 32358 loop1cycl 32377 |
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