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Theorem prsiga 30322
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 4864 . . 3 ∅ ∈ 𝒫 𝑂
2 pwidg 4206 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
3 prssi 4385 . . 3 ((∅ ∈ 𝒫 𝑂𝑂 ∈ 𝒫 𝑂) → {∅, 𝑂} ⊆ 𝒫 𝑂)
41, 2, 3sylancr 696 . 2 (𝑂𝑉 → {∅, 𝑂} ⊆ 𝒫 𝑂)
5 prid2g 4328 . . 3 (𝑂𝑉𝑂 ∈ {∅, 𝑂})
6 dif0 3983 . . . . 5 (𝑂 ∖ ∅) = 𝑂
76, 5syl5eqel 2734 . . . 4 (𝑂𝑉 → (𝑂 ∖ ∅) ∈ {∅, 𝑂})
8 difid 3981 . . . . 5 (𝑂𝑂) = ∅
9 0ex 4823 . . . . . . 7 ∅ ∈ V
109prid1 4329 . . . . . 6 ∅ ∈ {∅, 𝑂}
1110a1i 11 . . . . 5 (𝑂𝑉 → ∅ ∈ {∅, 𝑂})
128, 11syl5eqel 2734 . . . 4 (𝑂𝑉 → (𝑂𝑂) ∈ {∅, 𝑂})
13 difeq2 3755 . . . . . . 7 (𝑥 = ∅ → (𝑂𝑥) = (𝑂 ∖ ∅))
1413eleq1d 2715 . . . . . 6 (𝑥 = ∅ → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ ∅) ∈ {∅, 𝑂}))
15 difeq2 3755 . . . . . . 7 (𝑥 = 𝑂 → (𝑂𝑥) = (𝑂𝑂))
1615eleq1d 2715 . . . . . 6 (𝑥 = 𝑂 → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂𝑂) ∈ {∅, 𝑂}))
1714, 16ralprg 4266 . . . . 5 ((∅ ∈ V ∧ 𝑂𝑉) → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
189, 17mpan 706 . . . 4 (𝑂𝑉 → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
197, 12, 18mpbir2and 977 . . 3 (𝑂𝑉 → ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂})
20 uni0 4497 . . . . . . . . 9 ∅ = ∅
2120, 10eqeltri 2726 . . . . . . . 8 ∅ ∈ {∅, 𝑂}
229unisn 4483 . . . . . . . . 9 {∅} = ∅
2322, 10eqeltri 2726 . . . . . . . 8 {∅} ∈ {∅, 𝑂}
2421, 23pm3.2i 470 . . . . . . 7 ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})
25 snex 4938 . . . . . . . . 9 {∅} ∈ V
269, 25pm3.2i 470 . . . . . . . 8 (∅ ∈ V ∧ {∅} ∈ V)
27 unieq 4476 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
2827eleq1d 2715 . . . . . . . . 9 (𝑥 = ∅ → ( 𝑥 ∈ {∅, 𝑂} ↔ ∅ ∈ {∅, 𝑂}))
29 unieq 4476 . . . . . . . . . 10 (𝑥 = {∅} → 𝑥 = {∅})
3029eleq1d 2715 . . . . . . . . 9 (𝑥 = {∅} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅} ∈ {∅, 𝑂}))
3128, 30ralprg 4266 . . . . . . . 8 ((∅ ∈ V ∧ {∅} ∈ V) → (∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ↔ ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})))
3226, 31mp1i 13 . . . . . . 7 (𝑂𝑉 → (∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ↔ ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})))
3324, 32mpbiri 248 . . . . . 6 (𝑂𝑉 → ∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂})
34 unisng 4484 . . . . . . . 8 (𝑂𝑉 {𝑂} = 𝑂)
3534, 5eqeltrd 2730 . . . . . . 7 (𝑂𝑉 {𝑂} ∈ {∅, 𝑂})
36 uniprg 4482 . . . . . . . . . 10 ((∅ ∈ V ∧ 𝑂𝑉) → {∅, 𝑂} = (∅ ∪ 𝑂))
379, 36mpan 706 . . . . . . . . 9 (𝑂𝑉 {∅, 𝑂} = (∅ ∪ 𝑂))
38 uncom 3790 . . . . . . . . . 10 (∅ ∪ 𝑂) = (𝑂 ∪ ∅)
39 un0 4000 . . . . . . . . . 10 (𝑂 ∪ ∅) = 𝑂
4038, 39eqtri 2673 . . . . . . . . 9 (∅ ∪ 𝑂) = 𝑂
4137, 40syl6eq 2701 . . . . . . . 8 (𝑂𝑉 {∅, 𝑂} = 𝑂)
4241, 5eqeltrd 2730 . . . . . . 7 (𝑂𝑉 {∅, 𝑂} ∈ {∅, 𝑂})
43 snex 4938 . . . . . . . . 9 {𝑂} ∈ V
44 prex 4939 . . . . . . . . 9 {∅, 𝑂} ∈ V
4543, 44pm3.2i 470 . . . . . . . 8 ({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V)
46 unieq 4476 . . . . . . . . . 10 (𝑥 = {𝑂} → 𝑥 = {𝑂})
4746eleq1d 2715 . . . . . . . . 9 (𝑥 = {𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {𝑂} ∈ {∅, 𝑂}))
48 unieq 4476 . . . . . . . . . 10 (𝑥 = {∅, 𝑂} → 𝑥 = {∅, 𝑂})
4948eleq1d 2715 . . . . . . . . 9 (𝑥 = {∅, 𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅, 𝑂} ∈ {∅, 𝑂}))
5047, 49ralprg 4266 . . . . . . . 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 977 . . . . . 6 (𝑂𝑉 → ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂})
53 ralun 3828 . . . . . 6 ((∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂}) → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
5433, 52, 53syl2anc 694 . . . . 5 (𝑂𝑉 → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
55 pwpr 4462 . . . . . 6 𝒫 {∅, 𝑂} = ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})
5655raleqi 3172 . . . . 5 (∀𝑥 ∈ 𝒫 {∅, 𝑂} 𝑥 ∈ {∅, 𝑂} ↔ ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
5754, 56sylibr 224 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 {∅, 𝑂} 𝑥 ∈ {∅, 𝑂})
58 ax-1 6 . . . . 5 ( 𝑥 ∈ {∅, 𝑂} → (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂}))
5958ralimi 2981 . . . 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 1261 . 2 (𝑂𝑉 → (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂})))
62 issiga 30302 . . 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 699 1 (𝑂𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  {cpr 4212   cuni 4468   class class class wbr 4685  cfv 5926  ωcom 7107  cdom 7995  sigAlgebracsiga 30298
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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-siga 30299
This theorem is referenced by: (None)
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