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Mirrors > Home > MPE Home > Th. List > nbgrprc0 | Structured version Visualization version GIF version |
Description: The set of neighbors is empty if the graph 𝐺 or the vertex 𝑁 are proper classes. (Contributed by AV, 26-Oct-2020.) |
Ref | Expression |
---|---|
nbgrprc0 | ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nbgr 27123 | . . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) | |
2 | 1 | reldmmpo 7264 | . 2 ⊢ Rel dom NeighbVtx |
3 | 2 | ovprc 7173 | 1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 {csn 4525 {cpr 4527 ‘cfv 6324 (class class class)co 7135 Vtxcvtx 26789 Edgcedg 26840 NeighbVtx cnbgr 27122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-nbgr 27123 |
This theorem is referenced by: uhgrnbgr0nb 27144 nbgr0vtxlem 27145 |
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