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| Mirrors > Home > MPE Home > Th. List > acsmre | Structured version Visualization version GIF version | ||
| Description: Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
| Ref | Expression |
|---|---|
| acsmre | ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isacs 17694 | . 2 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 “ cima 5688 ⟶wf 6557 ‘cfv 6561 Fincfn 8985 Moorecmre 17625 ACScacs 17628 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-acs 17632 |
| This theorem is referenced by: acsfiel 17697 acsmred 17699 mreacs 17701 isacs3lem 18587 symggen 19488 odf1o1 19590 lsmmod 19693 gsumzsplit 19945 gsumzoppg 19962 gsumpt 19980 dmdprdd 20019 dprdfeq0 20042 dprdspan 20047 dprdres 20048 dprdss 20049 subgdmdprd 20054 subgdprd 20055 dprdsn 20056 dprd2dlem1 20061 dprd2da 20062 dmdprdsplit2lem 20065 ablfac1b 20090 pgpfac1lem1 20094 pgpfac1lem3 20097 pgpfac1lem4 20098 pgpfac1lem5 20099 pgpfaclem2 20102 isnacs2 42717 proot1mul 43206 proot1hash 43207 |
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