Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prsiga Structured version   Visualization version   GIF version

Theorem prsiga 31536
 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 5222 . . 3 ∅ ∈ 𝒫 𝑂
2 pwidg 4519 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
3 prssi 4714 . . 3 ((∅ ∈ 𝒫 𝑂𝑂 ∈ 𝒫 𝑂) → {∅, 𝑂} ⊆ 𝒫 𝑂)
41, 2, 3sylancr 590 . 2 (𝑂𝑉 → {∅, 𝑂} ⊆ 𝒫 𝑂)
5 prid2g 4657 . . 3 (𝑂𝑉𝑂 ∈ {∅, 𝑂})
6 dif0 4286 . . . . 5 (𝑂 ∖ ∅) = 𝑂
76, 5eqeltrid 2894 . . . 4 (𝑂𝑉 → (𝑂 ∖ ∅) ∈ {∅, 𝑂})
8 difid 4284 . . . . 5 (𝑂𝑂) = ∅
9 0ex 5176 . . . . . . 7 ∅ ∈ V
109prid1 4658 . . . . . 6 ∅ ∈ {∅, 𝑂}
1110a1i 11 . . . . 5 (𝑂𝑉 → ∅ ∈ {∅, 𝑂})
128, 11eqeltrid 2894 . . . 4 (𝑂𝑉 → (𝑂𝑂) ∈ {∅, 𝑂})
13 difeq2 4044 . . . . . . 7 (𝑥 = ∅ → (𝑂𝑥) = (𝑂 ∖ ∅))
1413eleq1d 2874 . . . . . 6 (𝑥 = ∅ → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ ∅) ∈ {∅, 𝑂}))
15 difeq2 4044 . . . . . . 7 (𝑥 = 𝑂 → (𝑂𝑥) = (𝑂𝑂))
1615eleq1d 2874 . . . . . 6 (𝑥 = 𝑂 → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂𝑂) ∈ {∅, 𝑂}))
1714, 16ralprg 4592 . . . . 5 ((∅ ∈ V ∧ 𝑂𝑉) → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
189, 17mpan 689 . . . 4 (𝑂𝑉 → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
197, 12, 18mpbir2and 712 . . 3 (𝑂𝑉 → ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂})
20 uni0 4829 . . . . . . . . 9 ∅ = ∅
2120, 10eqeltri 2886 . . . . . . . 8 ∅ ∈ {∅, 𝑂}
229unisn 4821 . . . . . . . . 9 {∅} = ∅
2322, 10eqeltri 2886 . . . . . . . 8 {∅} ∈ {∅, 𝑂}
2421, 23pm3.2i 474 . . . . . . 7 ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})
25 snex 5298 . . . . . . . . 9 {∅} ∈ V
269, 25pm3.2i 474 . . . . . . . 8 (∅ ∈ V ∧ {∅} ∈ V)
27 unieq 4812 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
2827eleq1d 2874 . . . . . . . . 9 (𝑥 = ∅ → ( 𝑥 ∈ {∅, 𝑂} ↔ ∅ ∈ {∅, 𝑂}))
29 unieq 4812 . . . . . . . . . 10 (𝑥 = {∅} → 𝑥 = {∅})
3029eleq1d 2874 . . . . . . . . 9 (𝑥 = {∅} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅} ∈ {∅, 𝑂}))
3128, 30ralprg 4592 . . . . . . . 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 4820 . . . . . . . 8 (𝑂𝑉 {𝑂} = 𝑂)
3534, 5eqeltrd 2890 . . . . . . 7 (𝑂𝑉 {𝑂} ∈ {∅, 𝑂})
36 uniprg 4819 . . . . . . . . . 10 ((∅ ∈ V ∧ 𝑂𝑉) → {∅, 𝑂} = (∅ ∪ 𝑂))
379, 36mpan 689 . . . . . . . . 9 (𝑂𝑉 {∅, 𝑂} = (∅ ∪ 𝑂))
38 uncom 4080 . . . . . . . . . 10 (∅ ∪ 𝑂) = (𝑂 ∪ ∅)
39 un0 4298 . . . . . . . . . 10 (𝑂 ∪ ∅) = 𝑂
4038, 39eqtri 2821 . . . . . . . . 9 (∅ ∪ 𝑂) = 𝑂
4137, 40eqtrdi 2849 . . . . . . . 8 (𝑂𝑉 {∅, 𝑂} = 𝑂)
4241, 5eqeltrd 2890 . . . . . . 7 (𝑂𝑉 {∅, 𝑂} ∈ {∅, 𝑂})
43 snex 5298 . . . . . . . . 9 {𝑂} ∈ V
44 prex 5299 . . . . . . . . 9 {∅, 𝑂} ∈ V
4543, 44pm3.2i 474 . . . . . . . 8 ({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V)
46 unieq 4812 . . . . . . . . . 10 (𝑥 = {𝑂} → 𝑥 = {𝑂})
4746eleq1d 2874 . . . . . . . . 9 (𝑥 = {𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {𝑂} ∈ {∅, 𝑂}))
48 unieq 4812 . . . . . . . . . 10 (𝑥 = {∅, 𝑂} → 𝑥 = {∅, 𝑂})
4948eleq1d 2874 . . . . . . . . 9 (𝑥 = {∅, 𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅, 𝑂} ∈ {∅, 𝑂}))
5047, 49ralprg 4592 . . . . . . . 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 712 . . . . . 6 (𝑂𝑉 → ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂})
53 ralun 4119 . . . . . 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 4795 . . . . . 6 𝒫 {∅, 𝑂} = ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})
5655raleqi 3362 . . . . 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 3128 . . . 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 1125 . 2 (𝑂𝑉 → (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂})))
62 issiga 31517 . . 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 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525  {cpr 4527  ∪ cuni 4801   class class class wbr 5031  ‘cfv 6327  ωcom 7567   ≼ cdom 8497  sigAlgebracsiga 31513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6286  df-fun 6329  df-fv 6335  df-siga 31514 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator