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Theorem clddisj 48801
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 48800 with elssuni 4937 replaced by the combination of cldss 23037 and eqid 2737. (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 3967 . 2 (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍))
2 simpl 482 . . . . 5 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = ( 𝐽𝑋))
3 cldrcl 23034 . . . . . . 7 (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4 clduni 48798 . . . . . . . 8 (𝐽 ∈ Top → (Clsd‘𝐽) = 𝐽)
54difeq1d 4125 . . . . . . 7 (𝐽 ∈ Top → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
63, 5syl 17 . . . . . 6 (𝑌 ∈ (Clsd‘𝐽) → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
76adantl 481 . . . . 5 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
82, 7eqtr4d 2780 . . . 4 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = ( (Clsd‘𝐽) ∖ 𝑋))
9 opndisj 48800 . . . . . 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 48715 . . . 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, 13bitrid 283 1 (𝑍 = ( 𝐽𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  cin 3950  c0 4333  𝒫 cpw 4600   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-mre 17629  df-top 22900  df-topon 22917  df-cld 23027
This theorem is referenced by: (None)
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