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Mirrors > Home > MPE Home > Th. List > nbgrprc0 | Structured version Visualization version Unicode version |
Description: The set of neighbors is empty if the graph or the vertex are proper classes. (Contributed by AV, 26-Oct-2020.) |
Ref | Expression |
---|---|
nbgrprc0 | NeighbVtx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nbgr 26228 | . . 3 NeighbVtx Vtx Vtx Edg | |
2 | 1 | reldmmpt2 6771 | . 2 NeighbVtx |
3 | 2 | ovprc 6683 | 1 NeighbVtx |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 wrex 2913 crab 2916 cvv 3200 cdif 3571 wss 3574 c0 3915 csn 4177 cpr 4179 cfv 5888 (class class class)co 6650 Vtxcvtx 25874 Edgcedg 25939 NeighbVtx cnbgr 26224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-nbgr 26228 |
This theorem is referenced by: uhgrnbgr0nb 26250 nbgr0vtxlem 26251 |
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