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

Step	Hyp	Ref	Expression
1		f0 6322	. . 3 ⊢ ∅:∅⟶∅
2		dm0 5570	. . . 4 ⊢ dom ∅ = ∅
3		pw0 4560	. . . . . 6 ⊢ 𝒫 ∅ = {∅}
4	3	difeq1i 3950	. . . . 5 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
5		difid 4177	. . . . 5 ⊢ ({∅} ∖ {∅}) = ∅
6	4, 5	eqtri 2848	. . . 4 ⊢ (𝒫 ∅ ∖ {∅}) = ∅
7	2, 6	feq23i 6271	. . 3 ⊢ (∅:dom ∅⟶(𝒫 ∅ ∖ {∅}) ↔ ∅:∅⟶∅)
8	1, 7	mpbir 223	. 2 ⊢ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})
9		0ex 5013	. . 3 ⊢ ∅ ∈ V
10		vtxval0 26336	. . . . 5 ⊢ (Vtx‘∅) = ∅
11	10	eqcomi 2833	. . . 4 ⊢ ∅ = (Vtx‘∅)
12		iedgval0 26337	. . . . 5 ⊢ (iEdg‘∅) = ∅
13	12	eqcomi 2833	. . . 4 ⊢ ∅ = (iEdg‘∅)
14	11, 13	isuhgr 26357	. . 3 ⊢ (∅ ∈ V → (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅})))
15	9, 14	ax-mp 5	. 2 ⊢ (∅ ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 ∅ ∖ {∅}))
16	8, 15	mpbir 223	1 ⊢ ∅ ∈ UHGraph

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)