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Mirrors > Home > MPE Home > Th. List > nbgrsym | Structured version Visualization version Unicode version |
Description: A vertex in a graph is a neighbor of a second vertex iff the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) |
Ref | Expression |
---|---|
nbgrsym | NeighbVtx NeighbVtx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 466 | . . . 4 Vtx Vtx Vtx Vtx | |
2 | necom 2847 | . . . 4 | |
3 | prcom 4267 | . . . . . 6 | |
4 | 3 | sseq1i 3629 | . . . . 5 |
5 | 4 | rexbii 3041 | . . . 4 Edg Edg |
6 | 1, 2, 5 | 3anbi123i 1251 | . . 3 Vtx Vtx Edg Vtx Vtx Edg |
7 | 6 | a1i 11 | . 2 Vtx Vtx Edg Vtx Vtx Edg |
8 | eqid 2622 | . . 3 Vtx Vtx | |
9 | eqid 2622 | . . 3 Edg Edg | |
10 | 8, 9 | nbgrel 26238 | . 2 NeighbVtx Vtx Vtx Edg |
11 | 8, 9 | nbgrel 26238 | . 2 NeighbVtx Vtx Vtx Edg |
12 | 7, 10, 11 | 3bitr4d 300 | 1 NeighbVtx NeighbVtx |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wcel 1990 wne 2794 wrex 2913 wss 3574 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 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 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-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-nbgr 26228 |
This theorem is referenced by: nbusgredgeu0 26270 uvtxanbgrvtx 26293 cplgr3v 26331 frgrncvvdeqlem1 27163 frgrwopreglem4a 27174 |
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