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Mirrors > Home > MPE Home > Th. List > Mathboxes > acycgr0v | Structured version Visualization version GIF version |
Description: A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
Ref | Expression |
---|---|
acycgr0v.1 | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
acycgr0v | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ AcyclicGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br0 5080 | . . . . . 6 ⊢ ¬ 𝑓∅𝑝 | |
2 | df-cycls 27731 | . . . . . . . . . 10 ⊢ Cycles = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}) | |
3 | 2 | relmptopab 7414 | . . . . . . . . 9 ⊢ Rel (Cycles‘𝐺) |
4 | cycliswlk 27742 | . . . . . . . . . 10 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) | |
5 | df-br 5032 | . . . . . . . . . 10 ⊢ (𝑓(Cycles‘𝐺)𝑝 ↔ 〈𝑓, 𝑝〉 ∈ (Cycles‘𝐺)) | |
6 | df-br 5032 | . . . . . . . . . 10 ⊢ (𝑓(Walks‘𝐺)𝑝 ↔ 〈𝑓, 𝑝〉 ∈ (Walks‘𝐺)) | |
7 | 4, 5, 6 | 3imtr3i 294 | . . . . . . . . 9 ⊢ (〈𝑓, 𝑝〉 ∈ (Cycles‘𝐺) → 〈𝑓, 𝑝〉 ∈ (Walks‘𝐺)) |
8 | 3, 7 | relssi 5632 | . . . . . . . 8 ⊢ (Cycles‘𝐺) ⊆ (Walks‘𝐺) |
9 | acycgr0v.1 | . . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 9 | eqeq1i 2744 | . . . . . . . . 9 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
11 | g0wlk0 27596 | . . . . . . . . 9 ⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅) | |
12 | 10, 11 | sylbi 220 | . . . . . . . 8 ⊢ (𝑉 = ∅ → (Walks‘𝐺) = ∅) |
13 | 8, 12 | sseqtrid 3930 | . . . . . . 7 ⊢ (𝑉 = ∅ → (Cycles‘𝐺) ⊆ ∅) |
14 | ss0 4288 | . . . . . . 7 ⊢ ((Cycles‘𝐺) ⊆ ∅ → (Cycles‘𝐺) = ∅) | |
15 | breq 5033 | . . . . . . . 8 ⊢ ((Cycles‘𝐺) = ∅ → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝑓∅𝑝)) | |
16 | 15 | notbid 321 | . . . . . . 7 ⊢ ((Cycles‘𝐺) = ∅ → (¬ 𝑓(Cycles‘𝐺)𝑝 ↔ ¬ 𝑓∅𝑝)) |
17 | 13, 14, 16 | 3syl 18 | . . . . . 6 ⊢ (𝑉 = ∅ → (¬ 𝑓(Cycles‘𝐺)𝑝 ↔ ¬ 𝑓∅𝑝)) |
18 | 1, 17 | mpbiri 261 | . . . . 5 ⊢ (𝑉 = ∅ → ¬ 𝑓(Cycles‘𝐺)𝑝) |
19 | 18 | intnanrd 493 | . . . 4 ⊢ (𝑉 = ∅ → ¬ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
20 | 19 | nexdv 1943 | . . 3 ⊢ (𝑉 = ∅ → ¬ ∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
21 | 20 | nexdv 1943 | . 2 ⊢ (𝑉 = ∅ → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
22 | isacycgr 32681 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
23 | 22 | biimpar 481 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) → 𝐺 ∈ AcyclicGraph) |
24 | 21, 23 | sylan2 596 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ AcyclicGraph) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 ≠ wne 2935 Vcvv 3399 ⊆ wss 3844 ∅c0 4212 〈cop 4523 class class class wbr 5031 ‘cfv 6340 0cc0 10618 ♯chash 13785 Vtxcvtx 26944 Walkscwlks 27541 Pathscpths 27656 Cyclesccycls 27729 AcyclicGraphcacycgr 32678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-fzo 13128 df-hash 13786 df-word 13959 df-wlks 27544 df-trls 27637 df-pths 27660 df-cycls 27731 df-acycgr 32679 |
This theorem is referenced by: (None) |
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