|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > uhgr0vusgr | Structured version Visualization version GIF version | ||
| Description: The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| uhgr0vusgr | ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ UHGraph) | |
| 2 | eqid 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 3 | eqid 2737 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
| 4 | 2, 3 | uhgr0v0e 29255 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (Edg‘𝐺) = ∅) | 
| 5 | uhgriedg0edg0 29144 | . . . 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 29253 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 UHGraphcuhgr 29073 USGraphcusgr 29166 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fv 6569 df-edg 29065 df-uhgr 29075 df-usgr 29168 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |