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Mirrors > Home > MPE Home > Th. List > Mathboxes > acycgrcycl | Structured version Visualization version GIF version |
Description: Any cycle in an acyclic graph is trivial (i.e. has one vertex and no edges). (Contributed by BTernaryTau, 12-Oct-2023.) |
Ref | Expression |
---|---|
acycgrcycl | ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝐹 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycliswlk 27577 | . . . . . . . 8 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
2 | wlkv 27392 | . . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) |
4 | 3 | simp2d 1138 | . . . . . 6 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹 ∈ V) |
5 | 4 | adantl 484 | . . . . 5 ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝐹 ∈ V) |
6 | 3 | simp3d 1139 | . . . . . 6 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝑃 ∈ V) |
7 | 6 | adantl 484 | . . . . 5 ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝑃 ∈ V) |
8 | breq1 5062 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝐹(Cycles‘𝐺)𝑝)) | |
9 | eqeq1 2824 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓 = ∅ ↔ 𝐹 = ∅)) | |
10 | 8, 9 | imbi12d 347 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅) ↔ (𝐹(Cycles‘𝐺)𝑝 → 𝐹 = ∅))) |
11 | breq2 5063 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝐹(Cycles‘𝐺)𝑝 ↔ 𝐹(Cycles‘𝐺)𝑃)) | |
12 | 11 | imbi1d 344 | . . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝐹(Cycles‘𝐺)𝑝 → 𝐹 = ∅) ↔ (𝐹(Cycles‘𝐺)𝑃 → 𝐹 = ∅))) |
13 | 10, 12 | sylan9bb 512 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅) ↔ (𝐹(Cycles‘𝐺)𝑃 → 𝐹 = ∅))) |
14 | isacycgr1 32414 | . . . . . . . 8 ⊢ (𝐺 ∈ AcyclicGraph → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅))) | |
15 | 14 | ibi 269 | . . . . . . 7 ⊢ (𝐺 ∈ AcyclicGraph → ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)) |
16 | 15 | 19.21bbi 2188 | . . . . . 6 ⊢ (𝐺 ∈ AcyclicGraph → (𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)) |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)) |
18 | 5, 7, 13, 17 | vtocl2d 3554 | . . . 4 ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (𝐹(Cycles‘𝐺)𝑃 → 𝐹 = ∅)) |
19 | 18 | ex 415 | . . 3 ⊢ (𝐺 ∈ AcyclicGraph → (𝐹(Cycles‘𝐺)𝑃 → (𝐹(Cycles‘𝐺)𝑃 → 𝐹 = ∅))) |
20 | 19 | pm2.43d 53 | . 2 ⊢ (𝐺 ∈ AcyclicGraph → (𝐹(Cycles‘𝐺)𝑃 → 𝐹 = ∅)) |
21 | 20 | imp 409 | 1 ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝐹 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 ∀wal 1534 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 class class class wbr 5059 ‘cfv 6348 Walkscwlks 27376 Cyclesccycls 27564 AcyclicGraphcacycgr 32410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-wlks 27379 df-trls 27472 df-pths 27495 df-cycls 27566 df-acycgr 32411 |
This theorem is referenced by: pthacycspth 32425 |
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