Theorem clddisj 47184

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 47183 with elssuni 4934 replaced by the combination of cldss 22462 and eqid 2731. (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 3960	. 2 ⊢ (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍))
2		simpl 483	. . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ 𝐽 ∖ 𝑋))
3		cldrcl 22459	. . . . . . 7 ⊢ (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4		clduni 47181	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
5	4	difeq1d 4117	. . . . . . 7 ⊢ (𝐽 ∈ Top → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋))
6	3, 5	syl 17	. . . . . 6 ⊢ (𝑌 ∈ (Clsd‘𝐽) → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋))
7	6	adantl 482	. . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋))
8	2, 7	eqtr4d 2774	. . . 4 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋))
9		opndisj 47183	. . . . . 6 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
10	1, 9	bitr3id 284	. . . . 5 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
11	10	pm5.32dra 47128	. . . 4 ⊢ ((𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋 ∩ 𝑌) = ∅))
12	8, 11	sylancom 588	. . 3 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋 ∩ 𝑌) = ∅))
13	12	pm5.32da 579	. 2 ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
14	1, 13	bitrid 282	1 ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∖ cdif 3941   ∩ cin 3943  ∅c0 4318  𝒫 cpw 4596  ∪ cuni 4901  ‘cfv 6532  Topctop 22324  Clsdccld 22449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fn 6535  df-fv 6540  df-mre 17512  df-top 22325  df-topon 22342  df-cld 22452

This theorem is referenced by: (None)