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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 28675 | . 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
2 | ibar 530 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))) | |
3 | iscusgrvtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | cplgruvtxb 28670 | . . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6544 Vtxcvtx 28256 USGraphcusgr 28409 UnivVtxcuvtx 28642 ComplGraphccplgr 28666 ComplUSGraphccusgr 28667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-cplgr 28668 df-cusgr 28669 |
This theorem is referenced by: vdiscusgrb 28787 vdiscusgr 28788 |
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