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Mirrors > Home > MPE Home > Th. List > nbgrasym | Structured version Visualization version GIF version |
Description: A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) |
Ref | Expression |
---|---|
nbgrasym | ⊢ (𝑉 USGrph 𝐸 → (𝑁 ∈ (〈𝑉, 𝐸〉 Neighbors 𝐾) ↔ 𝐾 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrav 25867 | . 2 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) | |
2 | 3ancoma 1038 | . . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)) | |
3 | prcom 4211 | . . . . . . 7 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾} | |
4 | 3 | eleq1i 2679 | . . . . . 6 ⊢ ({𝐾, 𝑁} ∈ ran 𝐸 ↔ {𝑁, 𝐾} ∈ ran 𝐸) |
5 | 4 | 3anbi3i 1248 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸)) |
6 | 2, 5 | bitri 263 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸)) |
7 | 6 | a1i 11 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸))) |
8 | nbgrael 25955 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ (〈𝑉, 𝐸〉 Neighbors 𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸))) | |
9 | nbgrael 25955 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐾 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁) ↔ (𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸))) | |
10 | 7, 8, 9 | 3bitr4d 299 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ (〈𝑉, 𝐸〉 Neighbors 𝐾) ↔ 𝐾 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁))) |
11 | 1, 10 | syl 17 | 1 ⊢ (𝑉 USGrph 𝐸 → (𝑁 ∈ (〈𝑉, 𝐸〉 Neighbors 𝐾) ↔ 𝐾 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 Vcvv 3173 {cpr 4127 〈cop 4131 class class class wbr 4583 ran crn 5039 (class class class)co 6549 USGrph cusg 25859 Neighbors cnbgra 25946 |
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-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-usgra 25862 df-nbgra 25949 |
This theorem is referenced by: uvtxnbgravtx 26023 |
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