Theorem 0vconngr 27053

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
0vconngr	|- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> G e. ConnGraph )

Proof of Theorem 0vconngr

Dummy variables f k n p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rzal 4073	. . 3 |- ( (Vtx ` G ) = (/) -> A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p )
2	1	adantl 482	. 2 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p )
3		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
4	3	isconngr 27049	. . 3 |- ( G e. W -> ( G e. ConnGraph <-> A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
5	4	adantr 481	. 2 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. ConnGraph <-> A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
6	2, 5	mpbird 247	1 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> G e. ConnGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  PathsOncpthson 26610  ConnGraphcconngr 27046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-conngr 27047

This theorem is referenced by:  1conngr  27054