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Mirrors > Home > MPE Home > Th. List > nbgrsym | Structured version Visualization version GIF version |
Description: In a graph, the neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Revised by AV, 12-Feb-2022.) |
Ref | Expression |
---|---|
nbgrsym | ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 464 | . . 3 ⊢ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ↔ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺))) | |
2 | necom 2988 | . . 3 ⊢ (𝑁 ≠ 𝐾 ↔ 𝐾 ≠ 𝑁) | |
3 | prcom 4624 | . . . . 5 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾} | |
4 | 3 | sseq1i 3906 | . . . 4 ⊢ ({𝐾, 𝑁} ⊆ 𝑒 ↔ {𝑁, 𝐾} ⊆ 𝑒) |
5 | 4 | rexbii 3162 | . . 3 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒) |
6 | 1, 2, 5 | 3anbi123i 1156 | . 2 ⊢ (((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 | eqid 2739 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | eqid 2739 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
9 | 7, 8 | nbgrel 27285 | . 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)) |
10 | 7, 8 | nbgrel 27285 | . 2 ⊢ (𝐾 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾 ≠ 𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)) |
11 | 6, 9, 10 | 3bitr4i 306 | 1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1088 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 ⊆ wss 3844 {cpr 4519 ‘cfv 6340 (class class class)co 7173 Vtxcvtx 26944 Edgcedg 26995 NeighbVtx cnbgr 27277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1st 7717 df-2nd 7718 df-nbgr 27278 |
This theorem is referenced by: nbusgredgeu0 27313 uvtxnbgrvtx 27338 cplgr3v 27380 frgrncvvdeqlem1 28239 frgrwopreglem4a 28250 numclwwlk1lem2foa 28294 |
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