Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nbgrssvwo2 | Structured version Visualization version GIF version |
Description: The neighbors of a vertex are a subset of all vertices except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) |
Ref | Expression |
---|---|
nbgrssovtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbgrssvwo2 | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrssovtx.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | nbgrssovtx 40586 | . . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁})) |
3 | df-nel 2783 | . . . . . 6 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁)) | |
4 | disjsn 4192 | . . . . . 6 ⊢ (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁)) | |
5 | 3, 4 | sylbb2 227 | . . . . 5 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → ((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅) |
6 | reldisj 3972 | . . . . 5 ⊢ ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))) | |
7 | 5, 6 | syl5ib 233 | . . . 4 ⊢ ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))) |
9 | 8 | imp 444 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})) |
10 | prcom 4211 | . . . 4 ⊢ {𝑀, 𝑁} = {𝑁, 𝑀} | |
11 | 10 | difeq2i 3687 | . . 3 ⊢ (𝑉 ∖ {𝑀, 𝑁}) = (𝑉 ∖ {𝑁, 𝑀}) |
12 | difpr 4275 | . . 3 ⊢ (𝑉 ∖ {𝑁, 𝑀}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀}) | |
13 | 11, 12 | eqtri 2632 | . 2 ⊢ (𝑉 ∖ {𝑀, 𝑁}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀}) |
14 | 9, 13 | syl6sseqr 3615 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 {cpr 4127 ‘cfv 5804 (class class class)co 6549 Vtxcvtx 25673 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-nel 2783 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: usgrnbssvwo2 40590 nbfusgrlevtxm2 40606 |
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