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Theorem incld 21049
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)) → (𝐴𝐵) ∈ (Clsd‘𝐽))

Proof of Theorem incld
StepHypRef Expression
1 intprg 4663 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} = (𝐴𝐵))
2 prnzg 4454 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → {𝐴, 𝐵} ≠ ∅)
32adantr 472 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ≠ ∅)
4 prssi 4498 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ⊆ (Clsd‘𝐽))
5 intcld 21046 . . 3 (({𝐴, 𝐵} ≠ ∅ ∧ {𝐴, 𝐵} ⊆ (Clsd‘𝐽)) → {𝐴, 𝐵} ∈ (Clsd‘𝐽))
63, 4, 5syl2anc 696 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → {𝐴, 𝐵} ∈ (Clsd‘𝐽))
71, 6eqeltrrd 2840 1 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wne 2932  cin 3714  wss 3715  c0 4058  {cpr 4323   cint 4627  cfv 6049  Clsdccld 21022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-top 20901  df-cld 21025
This theorem is referenced by:  riincld  21050  restcldr  21180  ordtcld3  21205  clsocv  23249  mblfinlem3  33761  mblfinlem4  33762
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