Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4459 | . . 3 ⊢ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅) | |
2 | vtxval0 26827 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
3 | 2 | raleqi 3416 | . . 3 ⊢ (∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) ↔ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘∅)) |
4 | 1, 3 | mpbir 233 | . 2 ⊢ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅) |
5 | 0ex 5214 | . . 3 ⊢ ∅ ∈ V | |
6 | eqid 2824 | . . . 4 ⊢ (Vtx‘∅) = (Vtx‘∅) | |
7 | 6 | iscplgr 27200 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅))) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (∅ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘∅)𝑣 ∈ (UnivVtx‘∅)) |
9 | 4, 8 | mpbir 233 | 1 ⊢ ∅ ∈ ComplGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ∅c0 4294 ‘cfv 6358 Vtxcvtx 26784 UnivVtxcuvtx 27170 ComplGraphccplgr 27194 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-slot 16490 df-base 16492 df-vtx 26786 df-uvtx 27171 df-cplgr 27196 |
This theorem is referenced by: cusgr0 27211 |
Copyright terms: Public domain | W3C validator |