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Theorem uhgr0 26370
Description: The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
Assertion
Ref Expression
uhgr0 ∅ ∈ UHGraph

Proof of Theorem uhgr0
StepHypRef Expression
1 f0 6322 . . 3 ∅:∅⟶∅
2 dm0 5570 . . . 4 dom ∅ = ∅
3 pw0 4560 . . . . . 6 𝒫 ∅ = {∅}
43difeq1i 3950 . . . . 5 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
5 difid 4177 . . . . 5 ({∅} ∖ {∅}) = ∅
64, 5eqtri 2848 . . . 4 (𝒫 ∅ ∖ {∅}) = ∅
72, 6feq23i 6271 . . 3 (∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) ↔ ∅:∅⟶∅)
81, 7mpbir 223 . 2 ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})
9 0ex 5013 . . 3 ∅ ∈ V
10 vtxval0 26336 . . . . 5 (Vtx‘∅) = ∅
1110eqcomi 2833 . . . 4 ∅ = (Vtx‘∅)
12 iedgval0 26337 . . . . 5 (iEdg‘∅) = ∅
1312eqcomi 2833 . . . 4 ∅ = (iEdg‘∅)
1411, 13isuhgr 26357 . . 3 (∅ ∈ V → (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})))
159, 14ax-mp 5 . 2 (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅}))
168, 15mpbir 223 1 ∅ ∈ UHGraph
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2166  Vcvv 3413  cdif 3794  c0 4143  𝒫 cpw 4377  {csn 4396  dom cdm 5341  wf 6118  cfv 6122  Vtxcvtx 26293  iEdgciedg 26294  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
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-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  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-slot 16225  df-base 16227  df-edgf 26287  df-vtx 26295  df-iedg 26296  df-uhgr 26355
This theorem is referenced by: (None)
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