Theorem clddisj 49562

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 49561 with elssuni 4905 replaced by the combination of cldss 23151 and eqid 2769. (Contributed by Zhi Wang, 6-Sep-2024.)

Assertion

Ref	Expression
clddisj	⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))

Proof of Theorem clddisj

Step	Hyp	Ref	Expression
1		elin 3929	. 2 ⊢ (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍))
2		simpl 487	. . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ 𝐽 ∖ 𝑋))
3		cldrcl 23148	. . . . . . 7 ⊢ (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4		clduni 49559	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
5	4	difeq1d 4088	. . . . . . 7 ⊢ (𝐽 ∈ Top → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋))
6	3, 5	syl 18	. . . . . 6 ⊢ (𝑌 ∈ (Clsd‘𝐽) → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋))
7	6	adantl 486	. . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋))
8	2, 7	eqtr4d 2807	. . . 4 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋))
9		opndisj 49561	. . . . . 6 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
10	1, 9	bitr3id 288	. . . . 5 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
11	10	pm5.32dra 49453	. . . 4 ⊢ ((𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋 ∩ 𝑌) = ∅))
12	8, 11	sylancom 599	. . 3 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋 ∩ 𝑌) = ∅))
13	12	pm5.32da 589	. 2 ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
14	1, 13	bitrid 286	1 ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910   ∩ cin 3912  ∅c0 4294  𝒫 cpw 4564  ∪ cuni 4873  ‘cfv 6534  Topctop 23015  Clsdccld 23138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fn 6537  df-fv 6542  df-mre 17634  df-top 23016  df-topon 23033  df-cld 23141

This theorem is referenced by: (None)