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Mirrors > Home > ILE Home > Th. List > cldss2 | 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 13793 | . . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋) |
3 | velpw 3584 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋) | |
4 | 2, 3 | sylibr 134 | . 2 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋) |
5 | 4 | ssriv 3161 | 1 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ⊆ wss 3131 𝒫 cpw 3577 ∪ cuni 3811 ‘cfv 5218 Clsdccld 13780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-top 13686 df-cld 13783 |
This theorem is referenced by: (None) |
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