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Theorem 0conngr 40106
Description: A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
Assertion
Ref Expression
0conngr  |-  (/)  e. ConnGraph

Proof of Theorem 0conngr
Dummy variables  f 
k  n  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3865 . 2  |-  A. k  e.  (/)  A. n  e.  (/)  E. f E. p  f ( k (PathsOn `  (/) ) n ) p
2 0ex 4528 . . 3  |-  (/)  e.  _V
3 vtxval0 39292 . . . . 5  |-  (Vtx `  (/) )  =  (/)
43eqcomi 2480 . . . 4  |-  (/)  =  (Vtx
`  (/) )
54isconngr 40103 . . 3  |-  ( (/)  e.  _V  ->  ( (/)  e. ConnGraph  <->  A. k  e.  (/)  A. n  e.  (/)  E. f E. p  f ( k (PathsOn `  (/) ) n ) p ) )
62, 5ax-mp 5 . 2  |-  ( (/)  e. ConnGraph  <->  A. k  e.  (/)  A. n  e.  (/)  E. f E. p  f ( k (PathsOn `  (/) ) n ) p )
71, 6mpbir 214 1  |-  (/)  e. ConnGraph
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   E.wex 1671    e. wcel 1904   A.wral 2756   _Vcvv 3031   (/)c0 3722   class class class wbr 4395   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251  PathsOncpthson 39909  ConnGraphcconngr 40100
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-ov 6311  df-slot 15203  df-base 15204  df-vtx 39253  df-conngr 40101
This theorem is referenced by: (None)
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