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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 27117 | . . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) | |
2 | 1 | reldmmpo 7287 | . 2 ⊢ Rel dom NeighbVtx |
3 | 2 | ovprc 7196 | 1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 {crab 3144 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 {csn 4569 {cpr 4571 ‘cfv 6357 (class class class)co 7158 Vtxcvtx 26783 Edgcedg 26834 NeighbVtx cnbgr 27116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dm 5567 df-iota 6316 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-nbgr 27117 |
This theorem is referenced by: uhgrnbgr0nb 27138 nbgr0vtxlem 27139 |
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