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Mirrors > Home > MPE Home > Th. List > 0conngr | Structured version Visualization version GIF version |
Description: A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
Ref | Expression |
---|---|
0conngr | ⊢ ∅ ∈ ConnGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4218 | . 2 ⊢ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝 | |
2 | 0ex 4925 | . . 3 ⊢ ∅ ∈ V | |
3 | vtxval0 26153 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
4 | 3 | eqcomi 2780 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
5 | 4 | isconngr 27370 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ ConnGraph ↔ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝)) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ (∅ ∈ ConnGraph ↔ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝) |
7 | 1, 6 | mpbir 221 | 1 ⊢ ∅ ∈ ConnGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∃wex 1852 ∈ wcel 2145 ∀wral 3061 Vcvv 3351 ∅c0 4064 class class class wbr 4787 ‘cfv 6032 (class class class)co 6794 Vtxcvtx 26096 PathsOncpthson 26846 ConnGraphcconngr 27367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3589 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-iota 5995 df-fun 6034 df-fv 6040 df-ov 6797 df-slot 16069 df-base 16071 df-vtx 26098 df-conngr 27368 |
This theorem is referenced by: (None) |
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