Theorem uhgr0vusgr 29222

Description: The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.)

Assertion

Ref	Expression
uhgr0vusgr	⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)

Proof of Theorem uhgr0vusgr

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
2		eqid 2733	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2733	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	2, 3	uhgr0v0e 29218	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (Edg‘𝐺) = ∅)
5		uhgriedg0edg0 29107	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
6	5	adantr 480	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
7	4, 6	mpbid 232	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
8	1, 7	usgr0e 29216	1 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∅c0 4282  ‘cfv 6486  Vtxcvtx 28976  iEdgciedg 28977  Edgcedg 29027  UHGraphcuhgr 29036  USGraphcusgr 29129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fv 6494  df-edg 29028  df-uhgr 29038  df-usgr 29131

This theorem is referenced by: (None)