| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > uvtxel | Structured version Visualization version GIF version | ||
| Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) (Revised by AV, 14-Feb-2022.) |
| Ref | Expression |
|---|---|
| uvtxel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| Ref | Expression |
|---|---|
| uvtxel | ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sneq 4636 | . . . 4 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁}) | |
| 2 | 1 | difeq2d 4126 | . . 3 ⊢ (𝑣 = 𝑁 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑁})) |
| 3 | oveq2 7439 | . . . 4 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁)) | |
| 4 | 3 | eleq2d 2827 | . . 3 ⊢ (𝑣 = 𝑁 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑁))) |
| 5 | 2, 4 | raleqbidv 3346 | . 2 ⊢ (𝑣 = 𝑁 → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))) |
| 6 | uvtxel.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 7 | 6 | uvtxval 29404 | . 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} |
| 8 | 5, 7 | elrab2 3695 | 1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 {csn 4626 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 NeighbVtx cnbgr 29349 UnivVtxcuvtx 29402 |
| 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 |
| 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-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-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-uvtx 29403 |
| This theorem is referenced by: uvtxisvtx 29406 vtxnbuvtx 29408 uvtx2vtx1edg 29415 uvtx2vtx1edgb 29416 uvtxnbgrb 29418 iscplgrnb 29433 cplgr1v 29447 cusgrexi 29460 structtocusgr 29463 |
| Copyright terms: Public domain | W3C validator |