Theorem edguhgr 26426

Description: An edge of a hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.)

Assertion

Ref	Expression
edguhgr	⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺))

Proof of Theorem edguhgr

Step	Hyp	Ref	Expression
1		uhgredgn0 26425	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
2	1	eldifad 3809	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺))

Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2166  ∅c0 4143  𝒫 cpw 4377  {csn 4396  ‘cfv 6122  Vtxcvtx 26293  Edgcedg 26344  UHGraphcuhgr 26353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fv 6130  df-edg 26345  df-uhgr 26355

This theorem is referenced by:  uhgredgrnv  26427  uhgrissubgr  26571  umgrres1lem  26606  nbuhgr  26639  nbuhgr2vtx1edgblem  26647  isomuspgrlem2c  42547