Theorem uvtxa0 26294

Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)

Hypothesis

Ref	Expression
uvtxael.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
uvtxa0	|- ( V = (/) -> (UnivVtx ` G ) = (/) )

Proof of Theorem uvtxa0

Dummy variables n v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxael.v	. . . . . 6 |- V = (Vtx ` G )
2	1	uvtxaval 26287	. . . . 5 |- ( G e. _V -> (UnivVtx ` G ) = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )
3	2	adantr 481	. . . 4 |- ( ( G e. _V /\ V = (/) ) -> (UnivVtx ` G ) = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )
4		rab0 3955	. . . . 5 |- { v e. (/) | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } = (/)
5		rabeq 3192	. . . . . . 7 |- ( V = (/) -> { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } = { v e. (/) | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )
6	5	eqeq1d 2624	. . . . . 6 |- ( V = (/) -> ( { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } = (/) <-> { v e. (/) | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } = (/) ) )
7	6	adantl 482	. . . . 5 |- ( ( G e. _V /\ V = (/) ) -> ( { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } = (/) <-> { v e. (/) | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } = (/) ) )
8	4, 7	mpbiri 248	. . . 4 |- ( ( G e. _V /\ V = (/) ) -> { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } = (/) )
9	3, 8	eqtrd 2656	. . 3 |- ( ( G e. _V /\ V = (/) ) -> (UnivVtx ` G ) = (/) )
10	9	ex 450	. 2 |- ( G e. _V -> ( V = (/) -> (UnivVtx ` G ) = (/) ) )
11		fvprc 6185	. . 3 |- ( -. G e. _V -> (UnivVtx ` G ) = (/) )
12	11	a1d 25	. 2 |- ( -. G e. _V -> ( V = (/) -> (UnivVtx ` G ) = (/) ) )
13	10, 12	pm2.61i 176	1 |- ( V = (/) -> (UnivVtx ` G ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874   NeighbVtx cnbgr 26224  UnivVtxcuvtxa 26225

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-uvtxa 26230

This theorem is referenced by: (None)