| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4979 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) | |
| 2 | prnzg 4776 | . . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → {𝐴, 𝐵} ≠ ∅) | |
| 3 | prssi 4819 | . . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ⊆ (Clsd‘𝐽)) | |
| 4 | intcld 23038 | . . 3 ⊢ (({𝐴, 𝐵} ≠ ∅ ∧ {𝐴, 𝐵} ⊆ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} ∈ (Clsd‘𝐽)) | |
| 5 | 2, 3, 4 | syl2an2r 685 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} ∈ (Clsd‘𝐽)) |
| 6 | 1, 5 | eqeltrrd 2841 | 1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2939 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 {cpr 4626 ∩ cint 4944 ‘cfv 6559 Clsdccld 23014 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6512 df-fun 6561 df-fn 6562 df-fv 6567 df-top 22890 df-cld 23017 |
| This theorem is referenced by: riincld 23042 restcldr 23172 ordtcld3 23197 clsocv 25274 mblfinlem3 37644 mblfinlem4 37645 iscnrm3rlem5 48814 |
| Copyright terms: Public domain | W3C validator |