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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 28942 | . 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
2 | ibar 527 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))) | |
3 | iscusgrvtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | cplgruvtxb 28937 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
5 | 2, 4 | bitr3d 280 | . 2 ⊢ (𝐺 ∈ USGraph → ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (UnivVtx‘𝐺) = 𝑉)) |
6 | 1, 5 | bitrid 282 | 1 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ‘cfv 6542 Vtxcvtx 28523 USGraphcusgr 28676 UnivVtxcuvtx 28909 ComplGraphccplgr 28933 ComplUSGraphccusgr 28934 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-cplgr 28935 df-cusgr 28936 |
This theorem is referenced by: vdiscusgrb 29054 vdiscusgr 29055 |
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