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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 28906 | . . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | eqid 2725 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2725 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
4 | 2, 3 | uhgrf 28919 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
5 | 4 | frnd 6725 | . . 3 ⊢ (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
6 | 1, 5 | eqsstrid 4021 | . 2 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 6 | sselda 3972 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ∖ cdif 3936 ∅c0 4318 𝒫 cpw 4598 {csn 4624 dom cdm 5672 ran crn 5673 ‘cfv 6543 Vtxcvtx 28853 iEdgciedg 28854 Edgcedg 28904 UHGraphcuhgr 28913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-edg 28905 df-uhgr 28915 |
This theorem is referenced by: edguhgr 28986 uhgredgss 28988 uhgrvd00 29392 lfuhgr2 34785 loop1cycl 34804 uspgrimprop 47283 |
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