Theorem 0grsubgr 27054

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 ∈ 𝑊 → ∅ SubGraph 𝐺)

Proof of Theorem 0grsubgr

Step	Hyp	Ref	Expression
1		0ss 4349	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		dm0 5784	. . . . 5 ⊢ dom ∅ = ∅
3	2	reseq2i 5844	. . . 4 ⊢ ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4		res0 5851	. . . 4 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
5	3, 4	eqtr2i 2845	. . 3 ⊢ ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6		0ss 4349	. . 3 ⊢ ∅ ⊆ 𝒫 ∅
7	1, 5, 6	3pm3.2i 1335	. 2 ⊢ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8		0ex 5203	. . 3 ⊢ ∅ ∈ V
9		vtxval0 26818	. . . . 5 ⊢ (Vtx‘∅) = ∅
10	9	eqcomi 2830	. . . 4 ⊢ ∅ = (Vtx‘∅)
11		eqid 2821	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		iedgval0 26819	. . . . 5 ⊢ (iEdg‘∅) = ∅
13	12	eqcomi 2830	. . . 4 ⊢ ∅ = (iEdg‘∅)
14		eqid 2821	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15		edgval 26828	. . . . 5 ⊢ (Edg‘∅) = ran (iEdg‘∅)
16	12	rneqi 5801	. . . . 5 ⊢ ran (iEdg‘∅) = ran ∅
17		rn0 5790	. . . . 5 ⊢ ran ∅ = ∅
18	15, 16, 17	3eqtrri 2849	. . . 4 ⊢ ∅ = (Edg‘∅)
19	10, 11, 13, 14, 18	issubgr 27047	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
20	8, 19	mpan2 689	. 2 ⊢ (𝐺 ∈ 𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
21	7, 20	mpbiri 260	1 ⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538   class class class wbr 5058  dom cdm 5549  ran crn 5550   ↾ cres 5551  ‘cfv 6349  Vtxcvtx 26775  iEdgciedg 26776  Edgcedg 26826   SubGraph csubgr 27043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-iota 6308  df-fun 6351  df-fv 6357  df-slot 16481  df-base 16483  df-edgf 26769  df-vtx 26777  df-iedg 26778  df-edg 26827  df-subgr 27044

This theorem is referenced by: (None)