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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 4300 | . . 3 ⊢ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅) | |
2 | vtxval0 26344 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
3 | 2 | raleqi 3354 | . . 3 ⊢ (∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) ↔ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅)) |
4 | 1, 3 | mpbir 223 | . 2 ⊢ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) |
5 | 0ex 5016 | . . 3 ⊢ ∅ ∈ V | |
6 | eqid 2825 | . . . 4 ⊢ (Vtx‘∅) = (Vtx‘∅) | |
7 | 6 | iscplgr 26720 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅))) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅)) |
9 | 4, 8 | mpbir 223 | 1 ⊢ ∅ ∈ ComplGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2164 ∀wral 3117 Vcvv 3414 ∅c0 4146 ‘cfv 6127 Vtxcvtx 26301 UnivVtxcuvtx 26690 ComplGraphccplgr 26714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-ov 6913 df-slot 16233 df-base 16235 df-vtx 26303 df-uvtx 26691 df-cplgr 26716 |
This theorem is referenced by: cusgr0 26731 |
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