Theorem clddisj 49391

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 49390 with elssuni 4882 replaced by the combination of cldss 23004 and eqid 2737. (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 3906	. 2 ⊢ (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍))
2		simpl 482	. . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ 𝐽 ∖ 𝑋))
3		cldrcl 23001	. . . . . . 7 ⊢ (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4		clduni 49388	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
5	4	difeq1d 4066	. . . . . . 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 481	. . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋))
8	2, 7	eqtr4d 2775	. . . 4 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋))
9		opndisj 49390	. . . . . 6 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
10	1, 9	bitr3id 285	. . . . 5 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
11	10	pm5.32dra 49282	. . . 4 ⊢ ((𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋 ∩ 𝑌) = ∅))
12	8, 11	sylancom 589	. . 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 283	1 ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887   ∩ cin 3889  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6492  Topctop 22868  Clsdccld 22991

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-mre 17539  df-top 22869  df-topon 22886  df-cld 22994

This theorem is referenced by: (None)