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Mirrors > Home > MPE Home > Th. List > edguhgr | Structured version Visualization version GIF version |
Description: An edge of a hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.) |
Ref | Expression |
---|---|
edguhgr | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgredgn0 28079 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) | |
2 | 1 | eldifad 3922 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∅c0 4282 𝒫 cpw 4560 {csn 4586 ‘cfv 6496 Vtxcvtx 27947 Edgcedg 27998 UHGraphcuhgr 28007 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-edg 27999 df-uhgr 28009 |
This theorem is referenced by: uhgredgrnv 28081 uhgrissubgr 28223 umgrres1lem 28258 nbuhgr 28291 nbuhgr2vtx1edgblem 28299 isomuspgrlem2c 46012 |
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