Theorem uhgr0v0e 26130

Description: The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)

Hypotheses

Ref	Expression
uhgr0v0e.v	|- V = (Vtx ` G )
uhgr0v0e.e	|- E = (Edg ` G )

Assertion

Ref	Expression
uhgr0v0e	|- ( ( G e. UHGraph /\ V = (/) ) -> E = (/) )

Proof of Theorem uhgr0v0e

Step	Hyp	Ref	Expression
1		uhgr0v0e.v	. . . . . 6 |- V = (Vtx ` G )
2	1	eqeq1i 2627	. . . . 5 |- ( V = (/) <-> (Vtx ` G ) = (/) )
3		uhgr0vb 25967	. . . . . . 7 |- ( ( G e. UHGraph /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph <-> (iEdg ` G ) = (/) ) )
4	3	biimpd 219	. . . . . 6 |- ( ( G e. UHGraph /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph -> (iEdg ` G ) = (/) ) )
5	4	ex 450	. . . . 5 |- ( G e. UHGraph -> ( (Vtx ` G ) = (/) -> ( G e. UHGraph -> (iEdg ` G ) = (/) ) ) )
6	2, 5	syl5bi 232	. . . 4 |- ( G e. UHGraph -> ( V = (/) -> ( G e. UHGraph -> (iEdg ` G ) = (/) ) ) )
7	6	pm2.43a 54	. . 3 |- ( G e. UHGraph -> ( V = (/) -> (iEdg ` G ) = (/) ) )
8	7	imp 445	. 2 |- ( ( G e. UHGraph /\ V = (/) ) -> (iEdg ` G ) = (/) )
9		uhgr0v0e.e	. . . . 5 |- E = (Edg ` G )
10	9	eqeq1i 2627	. . . 4 |- ( E = (/) <-> (Edg ` G ) = (/) )
11		uhgriedg0edg0 26022	. . . 4 |- ( G e. UHGraph -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
12	10, 11	syl5bb 272	. . 3 |- ( G e. UHGraph -> ( E = (/) <-> (iEdg ` G ) = (/) ) )
13	12	adantr 481	. 2 |- ( ( G e. UHGraph /\ V = (/) ) -> ( E = (/) <-> (iEdg ` G ) = (/) ) )
14	8, 13	mpbird 247	1 |- ( ( G e. UHGraph /\ V = (/) ) -> E = (/) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951

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  ax-un 6949

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-pw 4160  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-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-edg 25940  df-uhgr 25953

This theorem is referenced by:  uhgr0vsize0  26131  uhgr0vusgr  26134  fusgrfisbase  26220