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Theorem difopn 22889
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 4934 . . . . . 6 (𝐴𝐽𝐴 𝐽)
2 iscld.1 . . . . . 6 𝑋 = 𝐽
31, 2sseqtrrdi 4028 . . . . 5 (𝐴𝐽𝐴𝑋)
43adantr 480 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝑋)
5 df-ss 3960 . . . 4 (𝐴𝑋 ↔ (𝐴𝑋) = 𝐴)
64, 5sylib 217 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝑋) = 𝐴)
76difeq1d 4116 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) = (𝐴𝐵))
8 indif2 4265 . . 3 (𝐴 ∩ (𝑋𝐵)) = ((𝐴𝑋) ∖ 𝐵)
9 cldrcl 22881 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
109adantl 481 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11 simpl 482 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝐽)
122cldopn 22886 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
1312adantl 481 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝑋𝐵) ∈ 𝐽)
14 inopn 22752 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽 ∧ (𝑋𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
1510, 11, 13, 14syl3anc 1368 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
168, 15eqeltrrid 2832 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) ∈ 𝐽)
177, 16eqeltrrd 2828 1 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cdif 3940  cin 3942  wss 3943   cuni 4902  cfv 6536  Topctop 22746  Clsdccld 22871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-top 22747  df-cld 22874
This theorem is referenced by:  bcthlem5  25207  cldssbrsiga  33715  pibt2  36805  poimirlem30  37029  dirkercncflem2  45373  fourierdlem62  45437
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