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| Mirrors > Home > MPE Home > Th. List > cplgr0 | Structured version Visualization version GIF version | ||
| Description: The null graph (with no vertices and no edges) represented by the empty set is a complete graph. (Contributed by AV, 1-Nov-2020.) | 
| Ref | Expression | 
|---|---|
| cplgr0 | ⊢ ∅ ∈ ComplGraph | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ral0 4513 | . . 3 ⊢ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅) | |
| 2 | vtxval0 29056 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
| 3 | 2 | raleqi 3324 | . . 3 ⊢ (∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) ↔ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅)) | 
| 4 | 1, 3 | mpbir 231 | . 2 ⊢ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) | 
| 5 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 6 | eqid 2737 | . . . 4 ⊢ (Vtx‘∅) = (Vtx‘∅) | |
| 7 | 6 | iscplgr 29432 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅))) | 
| 8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅)) | 
| 9 | 4, 8 | mpbir 231 | 1 ⊢ ∅ ∈ ComplGraph | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∅c0 4333 ‘cfv 6561 Vtxcvtx 29013 UnivVtxcuvtx 29402 ComplGraphccplgr 29426 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-vtx 29015 df-uvtx 29403 df-cplgr 29428 | 
| This theorem is referenced by: cusgr0 29443 | 
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