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Theorem difopn 21931
Description: The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
difopn ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)

Proof of Theorem difopn
StepHypRef Expression
1 elssuni 4851 . . . . . 6 (𝐴𝐽𝐴 𝐽)
2 iscld.1 . . . . . 6 𝑋 = 𝐽
31, 2sseqtrrdi 3952 . . . . 5 (𝐴𝐽𝐴𝑋)
43adantr 484 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝑋)
5 df-ss 3883 . . . 4 (𝐴𝑋 ↔ (𝐴𝑋) = 𝐴)
64, 5sylib 221 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝑋) = 𝐴)
76difeq1d 4036 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) = (𝐴𝐵))
8 indif2 4185 . . 3 (𝐴 ∩ (𝑋𝐵)) = ((𝐴𝑋) ∖ 𝐵)
9 cldrcl 21923 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
109adantl 485 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11 simpl 486 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝐽)
122cldopn 21928 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
1312adantl 485 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝑋𝐵) ∈ 𝐽)
14 inopn 21796 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽 ∧ (𝑋𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
1510, 11, 13, 14syl3anc 1373 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
168, 15eqeltrrid 2843 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) ∈ 𝐽)
177, 16eqeltrrd 2839 1 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  cdif 3863  cin 3865  wss 3866   cuni 4819  cfv 6380  Topctop 21790  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-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:  bcthlem5  24225  cldssbrsiga  31867  pibt2  35325  poimirlem30  35544  dirkercncflem2  43320  fourierdlem62  43384
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