Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vtxdg0v Structured version   Visualization version   Unicode version

Theorem vtxdg0v 39698
Description: The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.)
Hypothesis
Ref Expression
vtxdgf.v  |-  V  =  (Vtx `  G )
Assertion
Ref Expression
vtxdg0v  |-  ( ( G  =  (/)  /\  U  e.  V )  ->  (
(VtxDeg `  G ) `  U )  =  0 )

Proof of Theorem vtxdg0v
StepHypRef Expression
1 vtxdgf.v . . . . 5  |-  V  =  (Vtx `  G )
21eleq2i 2541 . . . 4  |-  ( U  e.  V  <->  U  e.  (Vtx `  G ) )
3 fveq2 5879 . . . . . 6  |-  ( G  =  (/)  ->  (Vtx `  G )  =  (Vtx
`  (/) ) )
4 vtxval0 39292 . . . . . 6  |-  (Vtx `  (/) )  =  (/)
53, 4syl6eq 2521 . . . . 5  |-  ( G  =  (/)  ->  (Vtx `  G )  =  (/) )
65eleq2d 2534 . . . 4  |-  ( G  =  (/)  ->  ( U  e.  (Vtx `  G
)  <->  U  e.  (/) ) )
72, 6syl5bb 265 . . 3  |-  ( G  =  (/)  ->  ( U  e.  V  <->  U  e.  (/) ) )
8 noel 3726 . . . 4  |-  -.  U  e.  (/)
98pm2.21i 136 . . 3  |-  ( U  e.  (/)  ->  ( (VtxDeg `  G ) `  U
)  =  0 )
107, 9syl6bi 236 . 2  |-  ( G  =  (/)  ->  ( U  e.  V  ->  (
(VtxDeg `  G ) `  U )  =  0 ) )
1110imp 436 1  |-  ( ( G  =  (/)  /\  U  e.  V )  ->  (
(VtxDeg `  G ) `  U )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   (/)c0 3722   ` cfv 5589   0cc0 9557  Vtxcvtx 39251  VtxDegcvtxdg 39691
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
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  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-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  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-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-iota 5553  df-fun 5591  df-fv 5597  df-slot 15203  df-base 15204  df-vtx 39253
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator