Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nbgrssovtx Structured version   Visualization version   Unicode version

 Description: The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 39592. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
Hypothesis
Ref Expression
Assertion
Ref Expression

Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nbgrssovtx.v . . . . 5 Vtx
21nbgrisvtx 39591 . . . 4 NeighbVtx
3 nbgrnself2 39595 . . . . . . . . . 10 NeighbVtx
43adantr 472 . . . . . . . . 9 NeighbVtx
5 df-nel 2644 . . . . . . . . . 10 NeighbVtx NeighbVtx
6 neleq1 2748 . . . . . . . . . . 11 NeighbVtx NeighbVtx
76adantl 473 . . . . . . . . . 10 NeighbVtx NeighbVtx
85, 7syl5bbr 267 . . . . . . . . 9 NeighbVtx NeighbVtx
94, 8mpbird 240 . . . . . . . 8 NeighbVtx
109ex 441 . . . . . . 7 NeighbVtx
1110con2d 119 . . . . . 6 NeighbVtx
1211imp 436 . . . . 5 NeighbVtx
1312neqned 2650 . . . 4 NeighbVtx
14 eldifsn 4088 . . . 4
152, 13, 14sylanbrc 677 . . 3 NeighbVtx
1615ex 441 . 2 NeighbVtx
1716ssrdv 3424 1 NeighbVtx
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wa 376   wceq 1452   wcel 1904   wne 2641   wnel 2642   cdif 3387   wss 3390  csn 3959  cfv 5589  (class class class)co 6308  Vtxcvtx 39251   NeighbVtx cnbgr 39561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-nbgr 39565 This theorem is referenced by:  nbgrssvwo2  39597  usgrnbssovtx  39599  nbfusgrlevtxm1  39615  uvtxnbgr  39637  nbusgrvtxm1uvtx  39642  nbupgruvtxres  39644
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