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Theorem inelros 30567
Description: A ring of sets is closed under intersection. (Contributed by Thierry Arnoux, 19-Jul-2020.)
Hypothesis
Ref Expression
isros.1 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
Assertion
Ref Expression
inelros ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑠)   𝐵(𝑥,𝑦,𝑠)   𝑄(𝑥,𝑦,𝑠)   𝑂(𝑥,𝑦)

Proof of Theorem inelros
StepHypRef Expression
1 dfin4 4011 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
2 isros.1 . . . 4 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
32difelros 30566 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
42difelros 30566 . . 3 ((𝑆𝑄𝐴𝑆 ∧ (𝐴𝐵) ∈ 𝑆) → (𝐴 ∖ (𝐴𝐵)) ∈ 𝑆)
53, 4syld3an3 1516 . 2 ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴 ∖ (𝐴𝐵)) ∈ 𝑆)
61, 5syl5eqel 2844 1 ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051  {crab 3055  cdif 3713  cun 3714  cin 3715  c0 4059  𝒫 cpw 4303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730
This theorem is referenced by:  rossros  30574
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