Theorem 0conngr 27052

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.

Step	Hyp	Ref	Expression
1		ral0 4076	. 2 |- A. k e. (/) A. n e. (/) E. f E. p f ( k (PathsOn ` (/) ) n ) p
2		0ex 4790	. . 3 |- (/) e. _V
3		vtxval0 25931	. . . . 5 |- (Vtx ` (/) ) = (/)
4	3	eqcomi 2631	. . . 4 |- (/) = (Vtx ` (/) )
5	4	isconngr 27049	. . 3 |- ( (/) e. _V -> ( (/) e. ConnGraph <-> A. k e. (/) A. n e. (/) E. f E. p f ( k (PathsOn ` (/) ) n ) p ) )
6	2, 5	ax-mp 5	. 2 |- ( (/) e. ConnGraph <-> A. k e. (/) A. n e. (/) E. f E. p f ( k (PathsOn ` (/) ) n ) p )
7	1, 6	mpbir 221	1 |- (/) e. ConnGraph

Syntax hints:   <-> wb 196   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-mo 2475  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-slot 15861  df-base 15863  df-vtx 25876  df-conngr 27047

This theorem is referenced by: (None)