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Theorem inelsros 31430
Description: A semiring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
Hypothesis
Ref Expression
issros.1 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}
Assertion
Ref Expression
inelsros ((𝑆𝑁𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧
Allowed substitution hints:   𝐴(𝑡,𝑠)   𝐵(𝑥,𝑡,𝑠)   𝑆(𝑡)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem inelsros
StepHypRef Expression
1 simp2 1132 . . 3 ((𝑆𝑁𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 simp3 1133 . . 3 ((𝑆𝑁𝐴𝑆𝐵𝑆) → 𝐵𝑆)
3 issros.1 . . . . . 6 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}
43issros 31427 . . . . 5 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
54simp3bi 1142 . . . 4 (𝑆𝑁 → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
653ad2ant1 1128 . . 3 ((𝑆𝑁𝐴𝑆𝐵𝑆) → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
7 ineq1 4179 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
87eleq1d 2895 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝐴𝑦) ∈ 𝑆))
9 difeq1 4090 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
109eqeq1d 2821 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑦) = 𝑧 ↔ (𝐴𝑦) = 𝑧))
11103anbi3d 1436 . . . . . 6 (𝑥 = 𝐴 → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧)))
1211rexbidv 3295 . . . . 5 (𝑥 = 𝐴 → (∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧)))
138, 12anbi12d 632 . . . 4 (𝑥 = 𝐴 → (((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)) ↔ ((𝐴𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧))))
14 ineq2 4181 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
1514eleq1d 2895 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
16 difeq2 4091 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
1716eqeq1d 2821 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑦) = 𝑧 ↔ (𝐴𝐵) = 𝑧))
18173anbi3d 1436 . . . . . 6 (𝑦 = 𝐵 → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
1918rexbidv 3295 . . . . 5 (𝑦 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
2015, 19anbi12d 632 . . . 4 (𝑦 = 𝐵 → (((𝐴𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧)) ↔ ((𝐴𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧))))
2113, 20rspc2va 3632 . . 3 (((𝐴𝑆𝐵𝑆) ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))) → ((𝐴𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
221, 2, 6, 21syl21anc 835 . 2 ((𝑆𝑁𝐴𝑆𝐵𝑆) → ((𝐴𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
2322simpld 497 1 ((𝑆𝑁𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  wrex 3137  {crab 3140  cdif 3931  cin 3933  c0 4289  𝒫 cpw 4537   cuni 4830  Disj wdisj 5022  Fincfn 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950  df-pw 4539
This theorem is referenced by: (None)
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