Theorem 0grsubgr 26170

Description: The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)

Assertion

Ref	Expression
0grsubgr	|- ( G e. W -> (/) SubGraph G )

Proof of Theorem 0grsubgr

Step	Hyp	Ref	Expression
1		0ss 3972	. . 3 |- (/) C_ (Vtx ` G )
2		dm0 5339	. . . . 5 |- dom (/) = (/)
3	2	reseq2i 5393	. . . 4 |- ( (iEdg ` G ) |` dom (/) ) = ( (iEdg ` G ) |` (/) )
4		res0 5400	. . . 4 |- ( (iEdg ` G ) |` (/) ) = (/)
5	3, 4	eqtr2i 2645	. . 3 |- (/) = ( (iEdg ` G ) |` dom (/) )
6		0ss 3972	. . 3 |- (/) C_ ~P (/)
7	1, 5, 6	3pm3.2i 1239	. 2 |- ( (/) C_ (Vtx ` G ) /\ (/) = ( (iEdg ` G ) |` dom (/) ) /\ (/) C_ ~P (/) )
8		0ex 4790	. . 3 |- (/) e. _V
9		vtxval0 25931	. . . . 5 |- (Vtx ` (/) ) = (/)
10	9	eqcomi 2631	. . . 4 |- (/) = (Vtx ` (/) )
11		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
12		iedgval0 25932	. . . . 5 |- (iEdg ` (/) ) = (/)
13	12	eqcomi 2631	. . . 4 |- (/) = (iEdg ` (/) )
14		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
15		edgval 25941	. . . . 5 |- (Edg ` (/) ) = ran (iEdg ` (/) )
16	12	rneqi 5352	. . . . 5 |- ran (iEdg ` (/) ) = ran (/)
17		rn0 5377	. . . . 5 |- ran (/) = (/)
18	15, 16, 17	3eqtrri 2649	. . . 4 |- (/) = (Edg ` (/) )
19	10, 11, 13, 14, 18	issubgr 26163	. . 3 |- ( ( G e. W /\ (/) e. _V ) -> ( (/) SubGraph G <-> ( (/) C_ (Vtx ` G ) /\ (/) = ( (iEdg ` G ) |` dom (/) ) /\ (/) C_ ~P (/) ) ) )
20	8, 19	mpan2 707	. 2 |- ( G e. W -> ( (/) SubGraph G <-> ( (/) C_ (Vtx ` G ) /\ (/) = ( (iEdg ` G ) |` dom (/) ) /\ (/) C_ ~P (/) ) ) )
21	7, 20	mpbiri 248	1 |- ( G e. W -> (/) SubGraph G )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   SubGraph csubgr 26159

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-ne 2795  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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861  df-base 15863  df-edgf 25868  df-vtx 25876  df-iedg 25877  df-edg 25940  df-subgr 26160

This theorem is referenced by: (None)