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Mirrors > Home > MPE Home > Th. List > 0grsubgr | Structured version Visualization version GIF version |
Description: The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) |
Ref | Expression |
---|---|
0grsubgr | ⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4356 | . . 3 ⊢ ∅ ⊆ (Vtx‘𝐺) | |
2 | dm0 5876 | . . . . 5 ⊢ dom ∅ = ∅ | |
3 | 2 | reseq2i 5934 | . . . 4 ⊢ ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅) |
4 | res0 5941 | . . . 4 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅ | |
5 | 3, 4 | eqtr2i 2765 | . . 3 ⊢ ∅ = ((iEdg‘𝐺) ↾ dom ∅) |
6 | 0ss 4356 | . . 3 ⊢ ∅ ⊆ 𝒫 ∅ | |
7 | 1, 5, 6 | 3pm3.2i 1339 | . 2 ⊢ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅) |
8 | 0ex 5264 | . . 3 ⊢ ∅ ∈ V | |
9 | vtxval0 27988 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
10 | 9 | eqcomi 2745 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
11 | eqid 2736 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
12 | iedgval0 27989 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
13 | 12 | eqcomi 2745 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
14 | eqid 2736 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
15 | edgval 27998 | . . . . 5 ⊢ (Edg‘∅) = ran (iEdg‘∅) | |
16 | 12 | rneqi 5892 | . . . . 5 ⊢ ran (iEdg‘∅) = ran ∅ |
17 | rn0 5881 | . . . . 5 ⊢ ran ∅ = ∅ | |
18 | 15, 16, 17 | 3eqtrri 2769 | . . . 4 ⊢ ∅ = (Edg‘∅) |
19 | 10, 11, 13, 14, 18 | issubgr 28217 | . . 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 257 | 1 ⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 class class class wbr 5105 dom cdm 5633 ran crn 5634 ↾ cres 5635 ‘cfv 6496 Vtxcvtx 27945 iEdgciedg 27946 Edgcedg 27996 SubGraph csubgr 28213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-ltxr 11193 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-dec 12618 df-slot 17053 df-ndx 17065 df-base 17083 df-edgf 27936 df-vtx 27947 df-iedg 27948 df-edg 27997 df-subgr 28214 |
This theorem is referenced by: (None) |
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