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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 4974 | . 2 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
2 | dfss3 3953 | . . 3 ⊢ (𝐴 ⊆ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (Clsd‘𝐽)) | |
3 | iincld 21575 | . . 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 2914 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ∅c0 4288 ∩ cint 4867 ∩ ciin 4911 ‘cfv 6348 Clsdccld 21552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-top 21430 df-cld 21555 |
This theorem is referenced by: incld 21579 clscld 21583 cldmre 21614 |
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