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Mirrors > Home > MPE Home > Th. List > iscplgr | Structured version Visualization version GIF version |
Description: The property of being a complete graph. (Contributed by AV, 1-Nov-2020.) |
Ref | Expression |
---|---|
cplgruvtxb.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
iscplgr | ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cplgruvtxb.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | cplgruvtxb 27197 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
3 | eqss 3984 | . . 3 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺))) | |
4 | 1 | uvtxssvtx 27174 | . . . 4 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉 |
5 | dfss3 3958 | . . . . 5 ⊢ (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) | |
6 | 5 | anbi2i 624 | . . . 4 ⊢ (((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)) ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
7 | 4, 6 | mpbiran 707 | . . 3 ⊢ (((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) |
8 | 3, 7 | bitri 277 | . 2 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) |
9 | 2, 8 | syl6bb 289 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 ‘cfv 6357 Vtxcvtx 26783 UnivVtxcuvtx 27169 ComplGraphccplgr 27193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-uvtx 27170 df-cplgr 27195 |
This theorem is referenced by: iscplgrnb 27200 iscusgrvtx 27205 cplgr0 27209 cplgr0v 27211 cplgr1v 27214 cplgr2v 27216 cusgrexi 27227 structtocusgr 27230 cusgrres 27232 |
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