Theorem incld 21940

Description: The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
incld	⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽))

Proof of Theorem incld

Step	Hyp	Ref	Expression
1		intprg 4892	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
2		prnzg 4694	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → {𝐴, 𝐵} ≠ ∅)
3		prssi 4734	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ⊆ (Clsd‘𝐽))
4		intcld 21937	. . 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 2839	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2110   ≠ wne 2940   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  {cpr 4543  ∩ cint 4859  ‘cfv 6380  Clsdccld 21913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-top 21791  df-cld 21916

This theorem is referenced by:  riincld  21941  restcldr  22071  ordtcld3  22096  clsocv  24147  mblfinlem3  35553  mblfinlem4  35554  iscnrm3rlem5  45911