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Theorem difopn 21649
 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 4831 . . . . . 6 (𝐴𝐽𝐴 𝐽)
2 iscld.1 . . . . . 6 𝑋 = 𝐽
31, 2sseqtrrdi 3966 . . . . 5 (𝐴𝐽𝐴𝑋)
43adantr 484 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝑋)
5 df-ss 3898 . . . 4 (𝐴𝑋 ↔ (𝐴𝑋) = 𝐴)
64, 5sylib 221 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝑋) = 𝐴)
76difeq1d 4049 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) = (𝐴𝐵))
8 indif2 4197 . . 3 (𝐴 ∩ (𝑋𝐵)) = ((𝐴𝑋) ∖ 𝐵)
9 cldrcl 21641 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
109adantl 485 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11 simpl 486 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝐽)
122cldopn 21646 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
1312adantl 485 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝑋𝐵) ∈ 𝐽)
14 inopn 21514 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽 ∧ (𝑋𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
1510, 11, 13, 14syl3anc 1368 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
168, 15eqeltrrid 2895 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) ∈ 𝐽)
177, 16eqeltrrd 2891 1 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∪ cuni 4801  ‘cfv 6325  Topctop 21508  Clsdccld 21631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fn 6328  df-fv 6333  df-top 21509  df-cld 21634 This theorem is referenced by:  bcthlem5  23942  cldssbrsiga  31571  pibt2  34853  poimirlem30  35106  dirkercncflem2  42789  fourierdlem62  42853
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