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| Mirrors > Home > MPE Home > Th. List > cldss2 | Structured version Visualization version GIF version | ||
| Description: The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.) |
| Ref | Expression |
|---|---|
| iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| cldss2 | ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | 1 | cldss 23037 | . . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋) |
| 3 | velpw 4605 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋) | |
| 4 | 2, 3 | sylibr 234 | . 2 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋) |
| 5 | 4 | ssriv 3987 | 1 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ‘cfv 6561 Clsdccld 23024 |
| 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 ax-un 7755 |
| 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-fn 6564 df-fv 6569 df-top 22900 df-cld 23027 |
| This theorem is referenced by: cldmre 23086 cncls2 23281 fclscmp 24038 bcthlem5 25362 ubthlem1 30889 unicls 33902 clsf2 44139 |
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