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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 4988 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) | |
2 | prnzg 4785 | . . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → {𝐴, 𝐵} ≠ ∅) | |
3 | prssi 4828 | . . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ⊆ (Clsd‘𝐽)) | |
4 | intcld 23045 | . . 3 ⊢ (({𝐴, 𝐵} ≠ ∅ ∧ {𝐴, 𝐵} ⊆ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} ∈ (Clsd‘𝐽)) | |
5 | 2, 3, 4 | syl2an2r 684 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} ∈ (Clsd‘𝐽)) |
6 | 1, 5 | eqeltrrd 2838 | 1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2104 ≠ wne 2936 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {cpr 4632 ∩ cint 4953 ‘cfv 6558 Clsdccld 23021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6510 df-fun 6560 df-fn 6561 df-fv 6566 df-top 22897 df-cld 23024 |
This theorem is referenced by: riincld 23049 restcldr 23179 ordtcld3 23204 clsocv 25279 mblfinlem3 37606 mblfinlem4 37607 iscnrm3rlem5 48662 |
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