Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clddisj Structured version   Visualization version   GIF version

Theorem clddisj 45703
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 45702 with elssuni 4825 replaced by the combination of cldss 21773 and eqid 2738. (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 3857 . 2 (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍))
2 simpl 486 . . . . 5 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = ( 𝐽𝑋))
3 cldrcl 21770 . . . . . . 7 (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4 clduni 45700 . . . . . . . 8 (𝐽 ∈ Top → (Clsd‘𝐽) = 𝐽)
54difeq1d 4010 . . . . . . 7 (𝐽 ∈ Top → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
63, 5syl 17 . . . . . 6 (𝑌 ∈ (Clsd‘𝐽) → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
76adantl 485 . . . . 5 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ( (Clsd‘𝐽) ∖ 𝑋) = ( 𝐽𝑋))
82, 7eqtr4d 2776 . . . 4 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = ( (Clsd‘𝐽) ∖ 𝑋))
9 opndisj 45702 . . . . . 6 (𝑍 = ( (Clsd‘𝐽) ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
101, 9bitr3id 288 . . . . 5 (𝑍 = ( (Clsd‘𝐽) ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
1110pm5.32dra 45658 . . . 4 ((𝑍 = ( (Clsd‘𝐽) ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋𝑌) = ∅))
128, 11sylancom 591 . . 3 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋𝑌) = ∅))
1312pm5.32da 582 . 2 (𝑍 = ( 𝐽𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
141, 13syl5bb 286 1 (𝑍 = ( 𝐽𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  cdif 3838  cin 3840  c0 4209  𝒫 cpw 4485   cuni 4793  cfv 6333  Topctop 21637  Clsdccld 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-iin 4881  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fn 6336  df-fv 6341  df-mre 16953  df-top 21638  df-topon 21655  df-cld 21763
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator