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Theorem opndisj 45702
Description: Two ways of saying that two open sets are disjoint, if 𝐽 is a topology and 𝑋 is an open set. (Contributed by Zhi Wang, 6-Sep-2024.)
Assertion
Ref Expression
opndisj (𝑍 = ( 𝐽𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌𝐽 ∧ (𝑋𝑌) = ∅)))

Proof of Theorem opndisj
StepHypRef Expression
1 elpwg 4488 . . . 4 (𝑌𝐽 → (𝑌 ∈ 𝒫 𝑍𝑌𝑍))
2 sseq2 3901 . . . 4 (𝑍 = ( 𝐽𝑋) → (𝑌𝑍𝑌 ⊆ ( 𝐽𝑋)))
31, 2sylan9bbr 514 . . 3 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌𝐽) → (𝑌 ∈ 𝒫 𝑍𝑌 ⊆ ( 𝐽𝑋)))
43pm5.32da 582 . 2 (𝑍 = ( 𝐽𝑋) → ((𝑌𝐽𝑌 ∈ 𝒫 𝑍) ↔ (𝑌𝐽𝑌 ⊆ ( 𝐽𝑋))))
5 elin 3857 . 2 (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌𝐽𝑌 ∈ 𝒫 𝑍))
6 elssuni 4825 . . . 4 (𝑌𝐽𝑌 𝐽)
7 incom 4089 . . . . . 6 (𝑋𝑌) = (𝑌𝑋)
87eqeq1i 2743 . . . . 5 ((𝑋𝑌) = ∅ ↔ (𝑌𝑋) = ∅)
9 reldisj 4338 . . . . 5 (𝑌 𝐽 → ((𝑌𝑋) = ∅ ↔ 𝑌 ⊆ ( 𝐽𝑋)))
108, 9syl5bb 286 . . . 4 (𝑌 𝐽 → ((𝑋𝑌) = ∅ ↔ 𝑌 ⊆ ( 𝐽𝑋)))
116, 10syl 17 . . 3 (𝑌𝐽 → ((𝑋𝑌) = ∅ ↔ 𝑌 ⊆ ( 𝐽𝑋)))
1211pm5.32i 578 . 2 ((𝑌𝐽 ∧ (𝑋𝑌) = ∅) ↔ (𝑌𝐽𝑌 ⊆ ( 𝐽𝑋)))
134, 5, 123bitr4g 317 1 (𝑍 = ( 𝐽𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌𝐽 ∧ (𝑋𝑌) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  cdif 3838  cin 3840  wss 3841  c0 4209  𝒫 cpw 4485   cuni 4793
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-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3399  df-dif 3844  df-in 3848  df-ss 3858  df-nul 4210  df-pw 4487  df-uni 4794
This theorem is referenced by:  clddisj  45703
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