Theorem 0grsubgr 27062

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 4352	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		dm0 5792	. . . . 5 ⊢ dom ∅ = ∅
3	2	reseq2i 5852	. . . 4 ⊢ ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4		res0 5859	. . . 4 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
5	3, 4	eqtr2i 2847	. . 3 ⊢ ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6		0ss 4352	. . 3 ⊢ ∅ ⊆ 𝒫 ∅
7	1, 5, 6	3pm3.2i 1335	. 2 ⊢ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8		0ex 5213	. . 3 ⊢ ∅ ∈ V
9		vtxval0 26826	. . . . 5 ⊢ (Vtx‘∅) = ∅
10	9	eqcomi 2832	. . . 4 ⊢ ∅ = (Vtx‘∅)
11		eqid 2823	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		iedgval0 26827	. . . . 5 ⊢ (iEdg‘∅) = ∅
13	12	eqcomi 2832	. . . 4 ⊢ ∅ = (iEdg‘∅)
14		eqid 2823	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15		edgval 26836	. . . . 5 ⊢ (Edg‘∅) = ran (iEdg‘∅)
16	12	rneqi 5809	. . . . 5 ⊢ ran (iEdg‘∅) = ran ∅
17		rn0 5798	. . . . 5 ⊢ ran ∅ = ∅
18	15, 16, 17	3eqtrri 2851	. . . 4 ⊢ ∅ = (Edg‘∅)
19	10, 11, 13, 14, 18	issubgr 27055	. . 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 1537   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541   class class class wbr 5068  dom cdm 5557  ran crn 5558   ↾ cres 5559  ‘cfv 6357  Vtxcvtx 26783  iEdgciedg 26784  Edgcedg 26834   SubGraph csubgr 27051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365  df-slot 16489  df-base 16491  df-edgf 26777  df-vtx 26785  df-iedg 26786  df-edg 26835  df-subgr 27052

This theorem is referenced by: (None)