![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 16690 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | |
2 | 1 | simp1bi 1138 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ⊆ wss 3859 ∅c0 4211 𝒫 cpw 4453 ∩ cint 4782 ‘cfv 6225 Moorecmre 16682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-iota 6189 df-fun 6227 df-fv 6233 df-mre 16686 |
This theorem is referenced by: mress 16693 mrerintcl 16697 mreuni 16700 mremre 16704 isacs2 16753 mreacs 16758 isacs3lem 17605 dmdprdd 18838 dprdfeq0 18861 dprdss 18868 dprdz 18869 subgdmdprd 18873 subgdprd 18874 dprd2dlem1 18880 dprd2da 18881 dmdprdsplit2lem 18884 mretopd 21384 ismrc 38783 |
Copyright terms: Public domain | W3C validator |