Theorem uhgr0vusgr 29007

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 2726	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2726	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	2, 3	uhgr0v0e 29003	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (Edg‘𝐺) = ∅)
5		uhgriedg0edg0 28895	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
6	5	adantr 480	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
7	4, 6	mpbid 231	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
8	1, 7	usgr0e 29001	1 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∅c0 4317  ‘cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  UHGraphcuhgr 28824  USGraphcusgr 28917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fv 6545  df-edg 28816  df-uhgr 28826  df-usgr 28919

This theorem is referenced by: (None)