Theorem incld 22966

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 4954	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
2		prnzg 4751	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → {𝐴, 𝐵} ≠ ∅)
3		prssi 4794	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ⊆ (Clsd‘𝐽))
4		intcld 22963	. . 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 2834	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107   ≠ wne 2931   ∩ cin 3923   ⊆ wss 3924  ∅c0 4306  {cpr 4601  ∩ cint 4919  ‘cfv 6527  Clsdccld 22939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-int 4920  df-iun 4966  df-iin 4967  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6480  df-fun 6529  df-fn 6530  df-fv 6535  df-top 22817  df-cld 22942

This theorem is referenced by:  riincld  22967  restcldr  23097  ordtcld3  23122  clsocv  25187  mblfinlem3  37604  mblfinlem4  37605  iscnrm3rlem5  48797