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Theorem unelros 32136
Description: A ring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
Hypothesis
Ref Expression
isros.1 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
Assertion
Ref Expression
unelros ((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 unelros
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1136 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 simp3 1137 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → 𝐵𝑆)
3 isros.1 . . . . . 6 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
43isros 32133 . . . . 5 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
54simp3bi 1146 . . . 4 (𝑆𝑄 → ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
653ad2ant1 1132 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
7 uneq1 4091 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
87eleq1d 2823 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝑣) ∈ 𝑆 ↔ (𝐴𝑣) ∈ 𝑆))
9 difeq1 4051 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
109eleq1d 2823 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝑣) ∈ 𝑆 ↔ (𝐴𝑣) ∈ 𝑆))
118, 10anbi12d 631 . . . 4 (𝑢 = 𝐴 → (((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆) ↔ ((𝐴𝑣) ∈ 𝑆 ∧ (𝐴𝑣) ∈ 𝑆)))
12 uneq2 4092 . . . . . 6 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
1312eleq1d 2823 . . . . 5 (𝑣 = 𝐵 → ((𝐴𝑣) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
14 difeq2 4052 . . . . . 6 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
1514eleq1d 2823 . . . . 5 (𝑣 = 𝐵 → ((𝐴𝑣) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
1613, 15anbi12d 631 . . . 4 (𝑣 = 𝐵 → (((𝐴𝑣) ∈ 𝑆 ∧ (𝐴𝑣) ∈ 𝑆) ↔ ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆)))
1711, 16rspc2va 3572 . . 3 (((𝐴𝑆𝐵𝑆) ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)) → ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆))
181, 2, 6, 17syl21anc 835 . 2 ((𝑆𝑄𝐴𝑆𝐵𝑆) → ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆))
1918simpld 495 1 ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  {crab 3068  cdif 3885  cun 3886  c0 4258  𝒫 cpw 4535
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-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893
This theorem is referenced by:  fiunelros  32139
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