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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 4414 | . . 3 ⊢ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅) | |
2 | vtxval0 26832 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
3 | 2 | raleqi 3362 | . . 3 ⊢ (∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) ↔ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅)) |
4 | 1, 3 | mpbir 234 | . 2 ⊢ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) |
5 | 0ex 5175 | . . 3 ⊢ ∅ ∈ V | |
6 | eqid 2798 | . . . 4 ⊢ (Vtx‘∅) = (Vtx‘∅) | |
7 | 6 | iscplgr 27205 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅))) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅)) |
9 | 4, 8 | mpbir 234 | 1 ⊢ ∅ ∈ ComplGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∅c0 4243 ‘cfv 6324 Vtxcvtx 26789 UnivVtxcuvtx 27175 ComplGraphccplgr 27199 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-slot 16479 df-base 16481 df-vtx 26791 df-uvtx 27176 df-cplgr 27201 |
This theorem is referenced by: cusgr0 27216 |
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