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Mirrors > Home > MPE Home > Th. List > intcld | Structured version Visualization version GIF version |
Description: The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.) |
Ref | Expression |
---|---|
intcld | ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intiin 4993 | . 2 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
2 | dfss3 3913 | . . 3 ⊢ (𝐴 ⊆ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽)) | |
3 | iincld 22171 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽)) | |
4 | 2, 3 | sylan2b 593 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽)) |
5 | 1, 4 | eqeltrid 2844 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 ≠ wne 2944 ∀wral 3065 ⊆ wss 3891 ∅c0 4261 ∩ cint 4884 ∩ ciin 4930 ‘cfv 6430 Clsdccld 22148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-iota 6388 df-fun 6432 df-fn 6433 df-fv 6438 df-top 22024 df-cld 22151 |
This theorem is referenced by: incld 22175 clscld 22179 cldmre 22210 |
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