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| Mirrors > Home > MPE Home > Th. List > cusgruvtxb | Structured version Visualization version GIF version | ||
| Description: A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.) | 
| Ref | Expression | 
|---|---|
| iscusgrvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| Ref | Expression | 
|---|---|
| cusgruvtxb | ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iscusgr 29435 | . 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
| 2 | ibar 528 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))) | |
| 3 | iscusgrvtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 4 | 3 | cplgruvtxb 29430 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) | 
| 5 | 2, 4 | bitr3d 281 | . 2 ⊢ (𝐺 ∈ USGraph → ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (UnivVtx‘𝐺) = 𝑉)) | 
| 6 | 1, 5 | bitrid 283 | 1 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Vtxcvtx 29013 USGraphcusgr 29166 UnivVtxcuvtx 29402 ComplGraphccplgr 29426 ComplUSGraphccusgr 29427 | 
| 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-ext 2708 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-cplgr 29428 df-cusgr 29429 | 
| This theorem is referenced by: vdiscusgrb 29548 vdiscusgr 29549 | 
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