Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 21637 | . . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋) |
3 | velpw 4544 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋) | |
4 | 2, 3 | sylibr 236 | . 2 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋) |
5 | 4 | ssriv 3971 | 1 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 ‘cfv 6355 Clsdccld 21624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-top 21502 df-cld 21627 |
This theorem is referenced by: cldmre 21686 cncls2 21881 fclscmp 22638 bcthlem5 23931 ubthlem1 28647 unicls 31146 clsf2 40496 |
Copyright terms: Public domain | W3C validator |