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Theorem uhgr0 26869
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 6550 . . 3 ∅:∅⟶∅
2 dm0 5777 . . . 4 dom ∅ = ∅
3 pw0 4729 . . . . . 6 𝒫 ∅ = {∅}
43difeq1i 4081 . . . . 5 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
5 difid 4313 . . . . 5 ({∅} ∖ {∅}) = ∅
64, 5eqtri 2847 . . . 4 (𝒫 ∅ ∖ {∅}) = ∅
72, 6feq23i 6497 . . 3 (∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) ↔ ∅:∅⟶∅)
81, 7mpbir 234 . 2 ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})
9 0ex 5197 . . 3 ∅ ∈ V
10 vtxval0 26835 . . . . 5 (Vtx‘∅) = ∅
1110eqcomi 2833 . . . 4 ∅ = (Vtx‘∅)
12 iedgval0 26836 . . . . 5 (iEdg‘∅) = ∅
1312eqcomi 2833 . . . 4 ∅ = (iEdg‘∅)
1411, 13isuhgr 26856 . . 3 (∅ ∈ V → (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})))
159, 14ax-mp 5 . 2 (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅}))
168, 15mpbir 234 1 ∅ ∈ UHGraph
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  Vcvv 3480  cdif 3916  c0 4276  𝒫 cpw 4522  {csn 4550  dom cdm 5542  wf 6339  cfv 6343  Vtxcvtx 26792  iEdgciedg 26793  UHGraphcuhgr 26852
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-slot 16487  df-base 16489  df-edgf 26786  df-vtx 26794  df-iedg 26795  df-uhgr 26854
This theorem is referenced by: (None)
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