![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nbgrnself2 | Structured version Visualization version GIF version |
Description: A class ð is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.) |
Ref | Expression |
---|---|
nbgrnself2 | ⢠ð â (ðº NeighbVtx ð) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⢠(ð£ = ð â ð£ = ð) | |
2 | oveq2 7420 | . . . 4 ⢠(ð£ = ð â (ðº NeighbVtx ð£) = (ðº NeighbVtx ð)) | |
3 | 1, 2 | neleq12d 3050 | . . 3 ⢠(ð£ = ð â (ð£ â (ðº NeighbVtx ð£) â ð â (ðº NeighbVtx ð))) |
4 | eqid 2731 | . . . 4 ⢠(Vtxâðº) = (Vtxâðº) | |
5 | 4 | nbgrnself 28880 | . . 3 ⢠âð£ â (Vtxâðº)ð£ â (ðº NeighbVtx ð£) |
6 | 3, 5 | vtoclri 3577 | . 2 ⢠(ð â (Vtxâðº) â ð â (ðº NeighbVtx ð)) |
7 | 4 | nbgrisvtx 28862 | . . . 4 ⢠(ð â (ðº NeighbVtx ð) â ð â (Vtxâðº)) |
8 | 7 | con3i 154 | . . 3 ⢠(¬ ð â (Vtxâðº) â ¬ ð â (ðº NeighbVtx ð)) |
9 | df-nel 3046 | . . 3 ⢠(ð â (ðº NeighbVtx ð) â ¬ ð â (ðº NeighbVtx ð)) | |
10 | 8, 9 | sylibr 233 | . 2 ⢠(¬ ð â (Vtxâðº) â ð â (ðº NeighbVtx ð)) |
11 | 6, 10 | pm2.61i 182 | 1 ⢠ð â (ðº NeighbVtx ð) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1540 â wcel 2105 â wnel 3045 âcfv 6544 (class class class)co 7412 Vtxcvtx 28520 NeighbVtx cnbgr 28853 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-nbgr 28854 |
This theorem is referenced by: nbgrssovtx 28882 nb3grprlem2 28902 |
Copyright terms: Public domain | W3C validator |