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Mirrors > Home > MPE Home > Th. List > Mathboxes > isacycgr | Structured version Visualization version GIF version |
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
Ref | Expression |
---|---|
isacycgr | ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6663 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺)) | |
2 | 1 | breqd 5070 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝 ↔ 𝑓(Cycles‘𝐺)𝑝)) |
3 | 2 | anbi1d 631 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
4 | 3 | 2exbidv 1924 | . . 3 ⊢ (𝑔 = 𝐺 → (∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
5 | 4 | notbid 320 | . 2 ⊢ (𝑔 = 𝐺 → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
6 | df-acycgr 32411 | . 2 ⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)} | |
7 | 5, 6 | elab2g 3664 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3015 ∅c0 4284 class class class wbr 5059 ‘cfv 6348 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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-acycgr 32411 |
This theorem is referenced by: acycgr0v 32416 acycgr2v 32418 acycgrislfgr 32420 umgracycusgr 32422 cusgracyclt3v 32424 acycgrsubgr 32426 |
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