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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 5115 | . . . . . 6 ⊢ ¬ 𝑓∅𝑝 | |
2 | df-cycls 27568 | . . . . . . . . . 10 ⊢ Cycles = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}) | |
3 | 2 | relmptopab 7395 | . . . . . . . . 9 ⊢ Rel (Cycles‘𝐺) |
4 | cycliswlk 27579 | . . . . . . . . . 10 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) | |
5 | df-br 5067 | . . . . . . . . . 10 ⊢ (𝑓(Cycles‘𝐺)𝑝 ↔ 〈𝑓, 𝑝〉 ∈ (Cycles‘𝐺)) | |
6 | df-br 5067 | . . . . . . . . . 10 ⊢ (𝑓(Walks‘𝐺)𝑝 ↔ 〈𝑓, 𝑝〉 ∈ (Walks‘𝐺)) | |
7 | 4, 5, 6 | 3imtr3i 293 | . . . . . . . . 9 ⊢ (〈𝑓, 𝑝〉 ∈ (Cycles‘𝐺) → 〈𝑓, 𝑝〉 ∈ (Walks‘𝐺)) |
8 | 3, 7 | relssi 5660 | . . . . . . . 8 ⊢ (Cycles‘𝐺) ⊆ (Walks‘𝐺) |
9 | acycgr0v.1 | . . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 9 | eqeq1i 2826 | . . . . . . . . 9 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
11 | g0wlk0 27433 | . . . . . . . . 9 ⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅) | |
12 | 10, 11 | sylbi 219 | . . . . . . . 8 ⊢ (𝑉 = ∅ → (Walks‘𝐺) = ∅) |
13 | 8, 12 | sseqtrid 4019 | . . . . . . 7 ⊢ (𝑉 = ∅ → (Cycles‘𝐺) ⊆ ∅) |
14 | ss0 4352 | . . . . . . 7 ⊢ ((Cycles‘𝐺) ⊆ ∅ → (Cycles‘𝐺) = ∅) | |
15 | breq 5068 | . . . . . . . 8 ⊢ ((Cycles‘𝐺) = ∅ → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝑓∅𝑝)) | |
16 | 15 | notbid 320 | . . . . . . 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 260 | . . . . 5 ⊢ (𝑉 = ∅ → ¬ 𝑓(Cycles‘𝐺)𝑝) |
19 | 18 | intnanrd 492 | . . . 4 ⊢ (𝑉 = ∅ → ¬ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
20 | 19 | nexdv 1937 | . . 3 ⊢ (𝑉 = ∅ → ¬ ∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
21 | 20 | nexdv 1937 | . 2 ⊢ (𝑉 = ∅ → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
22 | isacycgr 32392 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
23 | 22 | biimpar 480 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) → 𝐺 ∈ AcyclicGraph) |
24 | 21, 23 | sylan2 594 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ AcyclicGraph) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 〈cop 4573 class class class wbr 5066 ‘cfv 6355 0cc0 10537 ♯chash 13691 Vtxcvtx 26781 Walkscwlks 27378 Pathscpths 27493 Cyclesccycls 27566 AcyclicGraphcacycgr 32389 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-wlks 27381 df-trls 27474 df-pths 27497 df-cycls 27568 df-acycgr 32390 |
This theorem is referenced by: (None) |
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