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| Mirrors > Home > MPE Home > Th. List > mresspw | Structured version Visualization version GIF version | ||
| Description: A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
| Ref | Expression |
|---|---|
| mresspw | ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ismre 17633 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
| 2 | 1 | simp1bi 1146 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∩ cint 4946 ‘cfv 6561 Moorecmre 17625 |
| 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-pow 5365 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-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-iota 6514 df-fun 6563 df-fv 6569 df-mre 17629 |
| This theorem is referenced by: mress 17636 mrerintcl 17640 mreuni 17643 mremre 17647 isacs2 17696 mreacs 17701 isacs3lem 18587 dmdprdd 20019 dprdfeq0 20042 dprdss 20049 dprdz 20050 subgdmdprd 20054 subgdprd 20055 dprd2dlem1 20061 dprd2da 20062 dmdprdsplit2lem 20065 mretopd 23100 ismrc 42712 |
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