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Mirrors > Home > MPE Home > Th. List > Mathboxes > clnbgrnvtx0 | Structured version Visualization version GIF version |
Description: If a class 𝑋 is not a vertex of a graph 𝐺, then it has an empty closed neighborhood in 𝐺. (Contributed by AV, 8-May-2025.) |
Ref | Expression |
---|---|
clnbgrel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
clnbgrnvtx0 | ⊢ (𝑋 ∉ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clnbgrel.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | csbfv 6957 | . . . . . 6 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺) | |
3 | 1, 2 | eqtr4i 2766 | . . . . 5 ⊢ 𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) |
4 | neleq2 3051 | . . . . 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 216 | . . 3 ⊢ (𝑋 ∉ 𝑉 → 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) |
7 | 6 | olcd 874 | . 2 ⊢ (𝑋 ∉ 𝑉 → (𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))) |
8 | df-clnbgr 47744 | . . 3 ⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})) | |
9 | 8 | mpoxneldm 8236 | . 2 ⊢ ((𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) → (𝐺 ClNeighbVtx 𝑋) = ∅) |
10 | 7, 9 | syl 17 | 1 ⊢ (𝑋 ∉ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1537 ∉ wnel 3044 ∃wrex 3068 {crab 3433 Vcvv 3478 ⦋csb 3908 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 {csn 4631 {cpr 4633 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 Edgcedg 29079 ClNeighbVtx cclnbgr 47743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-clnbgr 47744 |
This theorem is referenced by: clnbgr0vtx 47760 |
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