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Theorem clddisj 48700
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 48699 with elssuni 4942 replaced by the combination of cldss 23053 and eqid 2735. (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 3979 . 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 23050 . . . . . . 7 (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4 clduni 48697 . . . . . . . 8 (𝐽 ∈ Top → (Clsd‘𝐽) = 𝐽)
54difeq1d 4135 . . . . . . 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 2778 . . . 4 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = ( (Clsd‘𝐽) ∖ 𝑋))
9 opndisj 48699 . . . . . 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 48644 . . . 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 1537  wcel 2106  cdif 3960  cin 3962  c0 4339  𝒫 cpw 4605   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-mre 17631  df-top 22916  df-topon 22933  df-cld 23043
This theorem is referenced by: (None)
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