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Mirrors > Home > MPE Home > Th. List > nbgrnvtx0 | Structured version Visualization version GIF version |
Description: If a class 𝑋 is not a vertex of a graph 𝐺, then it has no neighbors in 𝐺. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
Ref | Expression |
---|---|
nbgrel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbgrnvtx0 | ⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrel.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | csbfv 6951 | . . . . . 6 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺) | |
3 | 1, 2 | eqtr4i 2757 | . . . . 5 ⊢ 𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) |
4 | neleq2 3043 | . . . . 5 ⊢ (𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) |
6 | 5 | biimpi 215 | . . 3 ⊢ (𝑋 ∉ 𝑉 → 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) |
7 | 6 | olcd 872 | . 2 ⊢ (𝑋 ∉ 𝑉 → (𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))) |
8 | df-nbgr 29269 | . . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) | |
9 | 8 | mpoxneldm 8227 | . 2 ⊢ ((𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) → (𝐺 NeighbVtx 𝑋) = ∅) |
10 | 7, 9 | syl 17 | 1 ⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1534 ∉ wnel 3036 ∃wrex 3060 {crab 3419 Vcvv 3462 ⦋csb 3892 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4325 {csn 4633 {cpr 4635 ‘cfv 6554 (class class class)co 7424 Vtxcvtx 28932 Edgcedg 28983 NeighbVtx cnbgr 29268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-nbgr 29269 |
This theorem is referenced by: nbuhgr 29279 nbumgr 29283 nbgr0vtx 29291 nbgr0edglem 29292 nbgr1vtx 29294 |
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