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Theorem difopn 12204
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 3734 . . . . . 6 (𝐴𝐽𝐴 𝐽)
2 iscld.1 . . . . . 6 𝑋 = 𝐽
31, 2sseqtrrdi 3116 . . . . 5 (𝐴𝐽𝐴𝑋)
43adantr 274 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝑋)
5 df-ss 3054 . . . 4 (𝐴𝑋 ↔ (𝐴𝑋) = 𝐴)
64, 5sylib 121 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝑋) = 𝐴)
76difeq1d 3163 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) = (𝐴𝐵))
8 indif2 3290 . . 3 (𝐴 ∩ (𝑋𝐵)) = ((𝐴𝑋) ∖ 𝐵)
9 cldrcl 12198 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
109adantl 275 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11 simpl 108 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → 𝐴𝐽)
122cldopn 12203 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
1312adantl 275 . . . 4 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝑋𝐵) ∈ 𝐽)
14 inopn 12097 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝐽 ∧ (𝑋𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
1510, 11, 13, 14syl3anc 1201 . . 3 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋𝐵)) ∈ 𝐽)
168, 15eqeltrrid 2205 . 2 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝑋) ∖ 𝐵) ∈ 𝐽)
177, 16eqeltrrd 2195 1 ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  cdif 3038  cin 3040  wss 3041   cuni 3706  cfv 5093  Topctop 12091  Clsdccld 12188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-top 12092  df-cld 12191
This theorem is referenced by: (None)
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