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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 ConnGraph

Proof of Theorem 0conngr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3865 . 2 PathsOn
2 0ex 4528 . . 3
3 vtxval0 39292 . . . . 5 Vtx
43eqcomi 2480 . . . 4 Vtx
54isconngr 40103 . . 3 ConnGraph PathsOn
62, 5ax-mp 5 . 2 ConnGraph PathsOn
71, 6mpbir 214 1 ConnGraph
 Colors of variables: wff setvar class Syntax hints:   wb 189  wex 1671   wcel 1904  wral 2756  cvv 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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