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Mirrors > Home > MPE Home > Th. List > incld | Structured version Visualization version GIF version |
Description: The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
incld | ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intprg 4940 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) | |
2 | prnzg 4737 | . . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → {𝐴, 𝐵} ≠ ∅) | |
3 | prssi 4779 | . . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ⊆ (Clsd‘𝐽)) | |
4 | intcld 22343 | . . 3 ⊢ (({𝐴, 𝐵} ≠ ∅ ∧ {𝐴, 𝐵} ⊆ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} ∈ (Clsd‘𝐽)) | |
5 | 2, 3, 4 | syl2an2r 683 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} ∈ (Clsd‘𝐽)) |
6 | 1, 5 | eqeltrrd 2839 | 1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ≠ wne 2941 ∩ cin 3907 ⊆ wss 3908 ∅c0 4280 {cpr 4586 ∩ cint 4905 ‘cfv 6493 Clsdccld 22319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fn 6496 df-fv 6501 df-top 22195 df-cld 22322 |
This theorem is referenced by: riincld 22347 restcldr 22477 ordtcld3 22502 clsocv 24566 mblfinlem3 36055 mblfinlem4 36056 iscnrm3rlem5 46878 |
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