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| Mirrors > Home > MPE Home > Th. List > iscusgrvtx | Structured version Visualization version GIF version | ||
| Description: A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.) |
| Ref | Expression |
|---|---|
| iscusgrvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| Ref | Expression |
|---|---|
| iscusgrvtx | ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscusgr 29350 | . 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
| 2 | iscusgrvtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 3 | 2 | iscplgr 29347 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
| 4 | 3 | pm5.32i 574 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
| 5 | 1, 4 | bitri 275 | 1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ‘cfv 6476 Vtxcvtx 28928 USGraphcusgr 29081 UnivVtxcuvtx 29317 ComplGraphccplgr 29341 ComplUSGraphccusgr 29342 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5231 ax-nul 5241 ax-pr 5367 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3393 df-v 3435 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5089 df-opab 5151 df-mpt 5170 df-id 5508 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-iota 6432 df-fun 6478 df-fv 6484 df-ov 7343 df-uvtx 29318 df-cplgr 29343 df-cusgr 29344 |
| This theorem is referenced by: (None) |
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