Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nbgrisvtx | Structured version Visualization version GIF version |
Description: Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
Ref | Expression |
---|---|
nbgrisvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbgrisvtx | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrisvtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2610 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | nbgrel 40564 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))) |
4 | simp1l 1078 | . . 3 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁 ∈ 𝑉) | |
5 | 3, 4 | syl6bi 242 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁 ∈ 𝑉)) |
6 | 5 | imp 444 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ⊆ wss 3540 {cpr 4127 ‘cfv 5804 (class class class)co 6549 Vtxcvtx 25673 Edgcedga 25792 NeighbVtx cnbgr 40550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-nbgr 40554 |
This theorem is referenced by: nbgrssvtx 40582 nbgrnself2 40585 nbgrssovtx 40586 frgrnbnb 41463 frgrncvvdeqlem3 41471 frgrncvvdeqlem4 41472 frgrncvvdeqlemC 41478 av-numclwwlkovf2ex 41517 av-numclwlk1lem2foa 41521 av-numclwlk1lem2fo 41525 |
Copyright terms: Public domain | W3C validator |