Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nbgrsym | Structured version Visualization version GIF version |
Description: A vertex in a graph is a neighbor of a second vertex iff the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) |
Ref | Expression |
---|---|
nbgrsym | ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 465 | . . . 4 ⊢ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ↔ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺))) | |
2 | necom 2835 | . . . 4 ⊢ (𝑁 ≠ 𝐾 ↔ 𝐾 ≠ 𝑁) | |
3 | prcom 4211 | . . . . . 6 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾} | |
4 | 3 | sseq1i 3592 | . . . . 5 ⊢ ({𝐾, 𝑁} ⊆ 𝑒 ↔ {𝑁, 𝐾} ⊆ 𝑒) |
5 | 4 | rexbii 3023 | . . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒) |
6 | 1, 2, 5 | 3anbi123i 1244 | . . 3 ⊢ (((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾 ≠ 𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)) |
7 | 6 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → (((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾 ≠ 𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))) |
8 | eqid 2610 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
9 | eqid 2610 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
10 | 8, 9 | nbgrel 40564 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))) |
11 | 8, 9 | nbgrel 40564 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝐾 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾 ≠ 𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))) |
12 | 7, 10, 11 | 3bitr4d 299 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ 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: nbusgredgeu0 40596 uvtxanbgrvtx 40619 cplgr3v 40657 |
Copyright terms: Public domain | W3C validator |