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Mirrors > Home > MPE Home > Th. List > nbgrnvtx0 | Structured version Visualization version Unicode version |
Description: There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
Ref | Expression |
---|---|
nbgrel.v | Vtx |
Ref | Expression |
---|---|
nbgrnvtx0 | NeighbVtx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrel.v | . . . . . 6 Vtx | |
2 | csbfv 6233 | . . . . . 6 Vtx Vtx | |
3 | 1, 2 | eqtr4i 2647 | . . . . 5 Vtx |
4 | neleq2 2903 | . . . . 5 Vtx Vtx | |
5 | 3, 4 | ax-mp 5 | . . . 4 Vtx |
6 | 5 | biimpi 206 | . . 3 Vtx |
7 | 6 | olcd 408 | . 2 Vtx |
8 | df-nbgr 26228 | . . 3 NeighbVtx Vtx Vtx Edg | |
9 | 8 | mpt2xneldm 7338 | . 2 Vtx NeighbVtx |
10 | 7, 9 | syl 17 | 1 NeighbVtx |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wceq 1483 wnel 2897 wrex 2913 crab 2916 cvv 3200 csb 3533 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 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-nel 2898 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: nbuhgr 26239 nbumgr 26243 nbgr0vtxlem 26251 nbgr1vtx 26254 |
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