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Theorem unelros 30362
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 1082 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 simp3 1083 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → 𝐵𝑆)
3 isros.1 . . . . . 6 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
43isros 30359 . . . . 5 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
54simp3bi 1098 . . . 4 (𝑆𝑄 → ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
653ad2ant1 1102 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
7 uneq1 3793 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
87eleq1d 2715 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝑣) ∈ 𝑆 ↔ (𝐴𝑣) ∈ 𝑆))
9 difeq1 3754 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
109eleq1d 2715 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝑣) ∈ 𝑆 ↔ (𝐴𝑣) ∈ 𝑆))
118, 10anbi12d 747 . . . 4 (𝑢 = 𝐴 → (((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆) ↔ ((𝐴𝑣) ∈ 𝑆 ∧ (𝐴𝑣) ∈ 𝑆)))
12 uneq2 3794 . . . . . 6 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
1312eleq1d 2715 . . . . 5 (𝑣 = 𝐵 → ((𝐴𝑣) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
14 difeq2 3755 . . . . . 6 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
1514eleq1d 2715 . . . . 5 (𝑣 = 𝐵 → ((𝐴𝑣) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
1613, 15anbi12d 747 . . . 4 (𝑣 = 𝐵 → (((𝐴𝑣) ∈ 𝑆 ∧ (𝐴𝑣) ∈ 𝑆) ↔ ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆)))
1711, 16rspc2va 3354 . . 3 (((𝐴𝑆𝐵𝑆) ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)) → ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆))
181, 2, 6, 17syl21anc 1365 . 2 ((𝑆𝑄𝐴𝑆𝐵𝑆) → ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆))
1918simpld 474 1 ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  cun 3605  c0 3948  𝒫 cpw 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612
This theorem is referenced by:  fiunelros  30365
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