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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 4454 | . 2 ⊢ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝 | |
2 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
3 | vtxval0 26816 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
4 | 3 | eqcomi 2828 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
5 | 4 | isconngr 27960 | . . 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 233 | 1 ⊢ ∅ ∈ ConnGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃wex 1774 ∈ wcel 2108 ∀wral 3136 Vcvv 3493 ∅c0 4289 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 Vtxcvtx 26773 PathsOncpthson 27487 ConnGraphcconngr 27957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7151 df-slot 16479 df-base 16481 df-vtx 26775 df-conngr 27958 |
This theorem is referenced by: (None) |
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