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Theorem clddisj 46197
Description: Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 46196 with elssuni 4871 replaced by the combination of cldss 22180 and eqid 2738. (Contributed by Zhi Wang, 6-Sep-2024.)
Assertion
Ref Expression
clddisj (𝑍 = ( 𝐽𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))

Proof of Theorem clddisj
StepHypRef Expression
1 elin 3903 . 2 (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍))
2 simpl 483 . . . . 5 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = ( 𝐽𝑋))
3 cldrcl 22177 . . . . . . 7 (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4 clduni 46194 . . . . . . . 8 (𝐽 ∈ Top → (Clsd‘𝐽) = 𝐽)
54difeq1d 4056 . . . . . . 7 (𝐽 ∈ Top → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
63, 5syl 17 . . . . . 6 (𝑌 ∈ (Clsd‘𝐽) → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
76adantl 482 . . . . 5 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
82, 7eqtr4d 2781 . . . 4 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = ( (Clsd‘𝐽) ∖ 𝑋))
9 opndisj 46196 . . . . . 6 (𝑍 = ( (Clsd‘𝐽) ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
101, 9bitr3id 285 . . . . 5 (𝑍 = ( (Clsd‘𝐽) ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
1110pm5.32dra 46140 . . . 4 ((𝑍 = ( (Clsd‘𝐽) ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋𝑌) = ∅))
128, 11sylancom 588 . . 3 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋𝑌) = ∅))
1312pm5.32da 579 . 2 (𝑍 = ( 𝐽𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
141, 13syl5bb 283 1 (𝑍 = ( 𝐽𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  cdif 3884  cin 3886  c0 4256  𝒫 cpw 4533   cuni 4839  cfv 6433  Topctop 22042  Clsdccld 22167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-mre 17295  df-top 22043  df-topon 22060  df-cld 22170
This theorem is referenced by: (None)
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