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Theorem uhgr0 26189
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 6248 . . 3 ∅:∅⟶∅
2 dm0 5495 . . . 4 dom ∅ = ∅
3 pw0 4489 . . . . . 6 𝒫 ∅ = {∅}
43difeq1i 3868 . . . . 5 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
5 difid 4092 . . . . 5 ({∅} ∖ {∅}) = ∅
64, 5eqtri 2783 . . . 4 (𝒫 ∅ ∖ {∅}) = ∅
72, 6feq23i 6201 . . 3 (∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) ↔ ∅:∅⟶∅)
81, 7mpbir 221 . 2 ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})
9 0ex 4943 . . 3 ∅ ∈ V
10 vtxval0 26152 . . . . 5 (Vtx‘∅) = ∅
1110eqcomi 2770 . . . 4 ∅ = (Vtx‘∅)
12 iedgval0 26153 . . . . 5 (iEdg‘∅) = ∅
1312eqcomi 2770 . . . 4 ∅ = (iEdg‘∅)
1411, 13isuhgr 26176 . . 3 (∅ ∈ V → (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})))
159, 14ax-mp 5 . 2 (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅}))
168, 15mpbir 221 1 ∅ ∈ UHGraph
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2140  Vcvv 3341  cdif 3713  c0 4059  𝒫 cpw 4303  {csn 4322  dom cdm 5267  wf 6046  cfv 6050  Vtxcvtx 26095  iEdgciedg 26096  UHGraphcuhgr 26172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-slot 16084  df-base 16086  df-edgf 26089  df-vtx 26097  df-iedg 26098  df-uhgr 26174
This theorem is referenced by: (None)
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