Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 4028 | . . 3 ⊢ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅) | |
2 | vtxval0 25714 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
3 | 2 | raleqi 3119 | . . 3 ⊢ (∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) ↔ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅)) |
4 | 1, 3 | mpbir 220 | . 2 ⊢ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) |
5 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
6 | eqid 2610 | . . . 4 ⊢ (Vtx‘∅) = (Vtx‘∅) | |
7 | 6 | iscplgr 40636 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅))) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅)) |
9 | 4, 8 | mpbir 220 | 1 ⊢ ∅ ∈ ComplGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∅c0 3874 ‘cfv 5804 Vtxcvtx 25673 UnivVtxcuvtxa 40551 ComplGraphccplgr 40552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-slot 15699 df-base 15700 df-vtx 25675 df-cplgr 40557 |
This theorem is referenced by: cusgr0 40648 |
Copyright terms: Public domain | W3C validator |