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Theorem prsiga 31811
Description: The smallest possible sigma-algebra containing 𝑂. (Contributed by Thierry Arnoux, 13-Sep-2016.)
Assertion
Ref Expression
prsiga (𝑂𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))

Proof of Theorem prsiga
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0elpw 5247 . . 3 ∅ ∈ 𝒫 𝑂
2 pwidg 4535 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
3 prssi 4734 . . 3 ((∅ ∈ 𝒫 𝑂𝑂 ∈ 𝒫 𝑂) → {∅, 𝑂} ⊆ 𝒫 𝑂)
41, 2, 3sylancr 590 . 2 (𝑂𝑉 → {∅, 𝑂} ⊆ 𝒫 𝑂)
5 prid2g 4677 . . 3 (𝑂𝑉𝑂 ∈ {∅, 𝑂})
6 dif0 4287 . . . . 5 (𝑂 ∖ ∅) = 𝑂
76, 5eqeltrid 2842 . . . 4 (𝑂𝑉 → (𝑂 ∖ ∅) ∈ {∅, 𝑂})
8 difid 4285 . . . . 5 (𝑂𝑂) = ∅
9 0ex 5200 . . . . . . 7 ∅ ∈ V
109prid1 4678 . . . . . 6 ∅ ∈ {∅, 𝑂}
1110a1i 11 . . . . 5 (𝑂𝑉 → ∅ ∈ {∅, 𝑂})
128, 11eqeltrid 2842 . . . 4 (𝑂𝑉 → (𝑂𝑂) ∈ {∅, 𝑂})
13 difeq2 4031 . . . . . . 7 (𝑥 = ∅ → (𝑂𝑥) = (𝑂 ∖ ∅))
1413eleq1d 2822 . . . . . 6 (𝑥 = ∅ → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ ∅) ∈ {∅, 𝑂}))
15 difeq2 4031 . . . . . . 7 (𝑥 = 𝑂 → (𝑂𝑥) = (𝑂𝑂))
1615eleq1d 2822 . . . . . 6 (𝑥 = 𝑂 → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂𝑂) ∈ {∅, 𝑂}))
1714, 16ralprg 4610 . . . . 5 ((∅ ∈ V ∧ 𝑂𝑉) → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
189, 17mpan 690 . . . 4 (𝑂𝑉 → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
197, 12, 18mpbir2and 713 . . 3 (𝑂𝑉 → ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂})
20 uni0 4849 . . . . . . . . 9 ∅ = ∅
2120, 10eqeltri 2834 . . . . . . . 8 ∅ ∈ {∅, 𝑂}
229unisn 4841 . . . . . . . . 9 {∅} = ∅
2322, 10eqeltri 2834 . . . . . . . 8 {∅} ∈ {∅, 𝑂}
2421, 23pm3.2i 474 . . . . . . 7 ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})
25 snex 5324 . . . . . . . . 9 {∅} ∈ V
269, 25pm3.2i 474 . . . . . . . 8 (∅ ∈ V ∧ {∅} ∈ V)
27 unieq 4830 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
2827eleq1d 2822 . . . . . . . . 9 (𝑥 = ∅ → ( 𝑥 ∈ {∅, 𝑂} ↔ ∅ ∈ {∅, 𝑂}))
29 unieq 4830 . . . . . . . . . 10 (𝑥 = {∅} → 𝑥 = {∅})
3029eleq1d 2822 . . . . . . . . 9 (𝑥 = {∅} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅} ∈ {∅, 𝑂}))
3128, 30ralprg 4610 . . . . . . . 8 ((∅ ∈ V ∧ {∅} ∈ V) → (∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ↔ ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})))
3226, 31mp1i 13 . . . . . . 7 (𝑂𝑉 → (∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ↔ ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})))
3324, 32mpbiri 261 . . . . . 6 (𝑂𝑉 → ∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂})
34 unisng 4840 . . . . . . . 8 (𝑂𝑉 {𝑂} = 𝑂)
3534, 5eqeltrd 2838 . . . . . . 7 (𝑂𝑉 {𝑂} ∈ {∅, 𝑂})
36 uniprg 4836 . . . . . . . . . 10 ((∅ ∈ V ∧ 𝑂𝑉) → {∅, 𝑂} = (∅ ∪ 𝑂))
379, 36mpan 690 . . . . . . . . 9 (𝑂𝑉 {∅, 𝑂} = (∅ ∪ 𝑂))
38 uncom 4067 . . . . . . . . . 10 (∅ ∪ 𝑂) = (𝑂 ∪ ∅)
39 un0 4305 . . . . . . . . . 10 (𝑂 ∪ ∅) = 𝑂
4038, 39eqtri 2765 . . . . . . . . 9 (∅ ∪ 𝑂) = 𝑂
4137, 40eqtrdi 2794 . . . . . . . 8 (𝑂𝑉 {∅, 𝑂} = 𝑂)
4241, 5eqeltrd 2838 . . . . . . 7 (𝑂𝑉 {∅, 𝑂} ∈ {∅, 𝑂})
43 snex 5324 . . . . . . . . 9 {𝑂} ∈ V
44 prex 5325 . . . . . . . . 9 {∅, 𝑂} ∈ V
4543, 44pm3.2i 474 . . . . . . . 8 ({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V)
46 unieq 4830 . . . . . . . . . 10 (𝑥 = {𝑂} → 𝑥 = {𝑂})
4746eleq1d 2822 . . . . . . . . 9 (𝑥 = {𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {𝑂} ∈ {∅, 𝑂}))
48 unieq 4830 . . . . . . . . . 10 (𝑥 = {∅, 𝑂} → 𝑥 = {∅, 𝑂})
4948eleq1d 2822 . . . . . . . . 9 (𝑥 = {∅, 𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅, 𝑂} ∈ {∅, 𝑂}))
5047, 49ralprg 4610 . . . . . . . 8 (({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V) → (∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂} ↔ ( {𝑂} ∈ {∅, 𝑂} ∧ {∅, 𝑂} ∈ {∅, 𝑂})))
5145, 50mp1i 13 . . . . . . 7 (𝑂𝑉 → (∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂} ↔ ( {𝑂} ∈ {∅, 𝑂} ∧ {∅, 𝑂} ∈ {∅, 𝑂})))
5235, 42, 51mpbir2and 713 . . . . . 6 (𝑂𝑉 → ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂})
53 ralun 4106 . . . . . 6 ((∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂}) → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
5433, 52, 53syl2anc 587 . . . . 5 (𝑂𝑉 → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
55 pwpr 4813 . . . . . 6 𝒫 {∅, 𝑂} = ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})
5655raleqi 3323 . . . . 5 (∀𝑥 ∈ 𝒫 {∅, 𝑂} 𝑥 ∈ {∅, 𝑂} ↔ ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
5754, 56sylibr 237 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 {∅, 𝑂} 𝑥 ∈ {∅, 𝑂})
58 ax-1 6 . . . . 5 ( 𝑥 ∈ {∅, 𝑂} → (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂}))
5958ralimi 3083 . . . 4 (∀𝑥 ∈ 𝒫 {∅, 𝑂} 𝑥 ∈ {∅, 𝑂} → ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂}))
6057, 59syl 17 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂}))
615, 19, 603jca 1130 . 2 (𝑂𝑉 → (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂})))
62 issiga 31792 . . 3 ({∅, 𝑂} ∈ V → ({∅, 𝑂} ∈ (sigAlgebra‘𝑂) ↔ ({∅, 𝑂} ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂})))))
6344, 62ax-mp 5 . 2 ({∅, 𝑂} ∈ (sigAlgebra‘𝑂) ↔ ({∅, 𝑂} ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂}))))
644, 61, 63sylanbrc 586 1 (𝑂𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408  cdif 3863  cun 3864  wss 3866  c0 4237  𝒫 cpw 4513  {csn 4541  {cpr 4543   cuni 4819   class class class wbr 5053  cfv 6380  ωcom 7644  cdom 8624  sigAlgebracsiga 31788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-siga 31789
This theorem is referenced by: (None)
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