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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 4505 | . . 3 ⊢ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅) | |
2 | vtxval0 28771 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
3 | 2 | raleqi 3315 | . . 3 ⊢ (∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) ↔ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅)) |
4 | 1, 3 | mpbir 230 | . 2 ⊢ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) |
5 | 0ex 5298 | . . 3 ⊢ ∅ ∈ V | |
6 | eqid 2724 | . . . 4 ⊢ (Vtx‘∅) = (Vtx‘∅) | |
7 | 6 | iscplgr 29144 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅))) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅)) |
9 | 4, 8 | mpbir 230 | 1 ⊢ ∅ ∈ ComplGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ∅c0 4315 ‘cfv 6534 Vtxcvtx 28728 UnivVtxcuvtx 29114 ComplGraphccplgr 29138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-nn 12211 df-slot 17116 df-ndx 17128 df-base 17146 df-vtx 28730 df-uvtx 29115 df-cplgr 29140 |
This theorem is referenced by: cusgr0 29155 |
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