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Theorem opndisj 46196
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 4536 . . . 4 (𝑌𝐽 → (𝑌 ∈ 𝒫 𝑍𝑌𝑍))
2 sseq2 3947 . . . 4 (𝑍 = ( 𝐽𝑋) → (𝑌𝑍𝑌 ⊆ ( 𝐽𝑋)))
31, 2sylan9bbr 511 . . 3 ((𝑍 = ( 𝐽𝑋) ∧ 𝑌𝐽) → (𝑌 ∈ 𝒫 𝑍𝑌 ⊆ ( 𝐽𝑋)))
43pm5.32da 579 . 2 (𝑍 = ( 𝐽𝑋) → ((𝑌𝐽𝑌 ∈ 𝒫 𝑍) ↔ (𝑌𝐽𝑌 ⊆ ( 𝐽𝑋))))
5 elin 3903 . 2 (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌𝐽𝑌 ∈ 𝒫 𝑍))
6 elssuni 4871 . . . 4 (𝑌𝐽𝑌 𝐽)
7 incom 4135 . . . . . 6 (𝑋𝑌) = (𝑌𝑋)
87eqeq1i 2743 . . . . 5 ((𝑋𝑌) = ∅ ↔ (𝑌𝑋) = ∅)
9 reldisj 4385 . . . . 5 (𝑌 𝐽 → ((𝑌𝑋) = ∅ ↔ 𝑌 ⊆ ( 𝐽𝑋)))
108, 9syl5bb 283 . . . 4 (𝑌 𝐽 → ((𝑋𝑌) = ∅ ↔ 𝑌 ⊆ ( 𝐽𝑋)))
116, 10syl 17 . . 3 (𝑌𝐽 → ((𝑋𝑌) = ∅ ↔ 𝑌 ⊆ ( 𝐽𝑋)))
1211pm5.32i 575 . 2 ((𝑌𝐽 ∧ (𝑋𝑌) = ∅) ↔ (𝑌𝐽𝑌 ⊆ ( 𝐽𝑋)))
134, 5, 123bitr4g 314 1 (𝑍 = ( 𝐽𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌𝐽 ∧ (𝑋𝑌) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  cdif 3884  cin 3886  wss 3887  c0 4256  𝒫 cpw 4533   cuni 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-uni 4840
This theorem is referenced by:  clddisj  46197
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