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Theorem prsiga 30662
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 4994 . . 3 ∅ ∈ 𝒫 𝑂
2 pwidg 4332 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
3 prssi 4508 . . 3 ((∅ ∈ 𝒫 𝑂𝑂 ∈ 𝒫 𝑂) → {∅, 𝑂} ⊆ 𝒫 𝑂)
41, 2, 3sylancr 581 . 2 (𝑂𝑉 → {∅, 𝑂} ⊆ 𝒫 𝑂)
5 prid2g 4453 . . 3 (𝑂𝑉𝑂 ∈ {∅, 𝑂})
6 dif0 4117 . . . . 5 (𝑂 ∖ ∅) = 𝑂
76, 5syl5eqel 2848 . . . 4 (𝑂𝑉 → (𝑂 ∖ ∅) ∈ {∅, 𝑂})
8 difid 4115 . . . . 5 (𝑂𝑂) = ∅
9 0ex 4952 . . . . . . 7 ∅ ∈ V
109prid1 4454 . . . . . 6 ∅ ∈ {∅, 𝑂}
1110a1i 11 . . . . 5 (𝑂𝑉 → ∅ ∈ {∅, 𝑂})
128, 11syl5eqel 2848 . . . 4 (𝑂𝑉 → (𝑂𝑂) ∈ {∅, 𝑂})
13 difeq2 3886 . . . . . . 7 (𝑥 = ∅ → (𝑂𝑥) = (𝑂 ∖ ∅))
1413eleq1d 2829 . . . . . 6 (𝑥 = ∅ → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ ∅) ∈ {∅, 𝑂}))
15 difeq2 3886 . . . . . . 7 (𝑥 = 𝑂 → (𝑂𝑥) = (𝑂𝑂))
1615eleq1d 2829 . . . . . 6 (𝑥 = 𝑂 → ((𝑂𝑥) ∈ {∅, 𝑂} ↔ (𝑂𝑂) ∈ {∅, 𝑂}))
1714, 16ralprg 4392 . . . . 5 ((∅ ∈ V ∧ 𝑂𝑉) → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
189, 17mpan 681 . . . 4 (𝑂𝑉 → (∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂𝑂) ∈ {∅, 𝑂})))
197, 12, 18mpbir2and 704 . . 3 (𝑂𝑉 → ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂})
20 uni0 4625 . . . . . . . . 9 ∅ = ∅
2120, 10eqeltri 2840 . . . . . . . 8 ∅ ∈ {∅, 𝑂}
229unisn 4612 . . . . . . . . 9 {∅} = ∅
2322, 10eqeltri 2840 . . . . . . . 8 {∅} ∈ {∅, 𝑂}
2421, 23pm3.2i 462 . . . . . . 7 ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})
25 snex 5066 . . . . . . . . 9 {∅} ∈ V
269, 25pm3.2i 462 . . . . . . . 8 (∅ ∈ V ∧ {∅} ∈ V)
27 unieq 4604 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
2827eleq1d 2829 . . . . . . . . 9 (𝑥 = ∅ → ( 𝑥 ∈ {∅, 𝑂} ↔ ∅ ∈ {∅, 𝑂}))
29 unieq 4604 . . . . . . . . . 10 (𝑥 = {∅} → 𝑥 = {∅})
3029eleq1d 2829 . . . . . . . . 9 (𝑥 = {∅} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅} ∈ {∅, 𝑂}))
3128, 30ralprg 4392 . . . . . . . 8 ((∅ ∈ V ∧ {∅} ∈ V) → (∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ↔ ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})))
3226, 31mp1i 13 . . . . . . 7 (𝑂𝑉 → (∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ↔ ( ∅ ∈ {∅, 𝑂} ∧ {∅} ∈ {∅, 𝑂})))
3324, 32mpbiri 249 . . . . . 6 (𝑂𝑉 → ∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂})
34 unisng 4611 . . . . . . . 8 (𝑂𝑉 {𝑂} = 𝑂)
3534, 5eqeltrd 2844 . . . . . . 7 (𝑂𝑉 {𝑂} ∈ {∅, 𝑂})
36 uniprg 4610 . . . . . . . . . 10 ((∅ ∈ V ∧ 𝑂𝑉) → {∅, 𝑂} = (∅ ∪ 𝑂))
379, 36mpan 681 . . . . . . . . 9 (𝑂𝑉 {∅, 𝑂} = (∅ ∪ 𝑂))
38 uncom 3921 . . . . . . . . . 10 (∅ ∪ 𝑂) = (𝑂 ∪ ∅)
39 un0 4131 . . . . . . . . . 10 (𝑂 ∪ ∅) = 𝑂
4038, 39eqtri 2787 . . . . . . . . 9 (∅ ∪ 𝑂) = 𝑂
4137, 40syl6eq 2815 . . . . . . . 8 (𝑂𝑉 {∅, 𝑂} = 𝑂)
4241, 5eqeltrd 2844 . . . . . . 7 (𝑂𝑉 {∅, 𝑂} ∈ {∅, 𝑂})
43 snex 5066 . . . . . . . . 9 {𝑂} ∈ V
44 prex 5067 . . . . . . . . 9 {∅, 𝑂} ∈ V
4543, 44pm3.2i 462 . . . . . . . 8 ({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V)
46 unieq 4604 . . . . . . . . . 10 (𝑥 = {𝑂} → 𝑥 = {𝑂})
4746eleq1d 2829 . . . . . . . . 9 (𝑥 = {𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {𝑂} ∈ {∅, 𝑂}))
48 unieq 4604 . . . . . . . . . 10 (𝑥 = {∅, 𝑂} → 𝑥 = {∅, 𝑂})
4948eleq1d 2829 . . . . . . . . 9 (𝑥 = {∅, 𝑂} → ( 𝑥 ∈ {∅, 𝑂} ↔ {∅, 𝑂} ∈ {∅, 𝑂}))
5047, 49ralprg 4392 . . . . . . . 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 704 . . . . . 6 (𝑂𝑉 → ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂})
53 ralun 3959 . . . . . 6 ((∀𝑥 ∈ {∅, {∅}} 𝑥 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}} 𝑥 ∈ {∅, 𝑂}) → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
5433, 52, 53syl2anc 579 . . . . 5 (𝑂𝑉 → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
55 pwpr 4590 . . . . . 6 𝒫 {∅, 𝑂} = ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})
5655raleqi 3290 . . . . 5 (∀𝑥 ∈ 𝒫 {∅, 𝑂} 𝑥 ∈ {∅, 𝑂} ↔ ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}}) 𝑥 ∈ {∅, 𝑂})
5754, 56sylibr 225 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 {∅, 𝑂} 𝑥 ∈ {∅, 𝑂})
58 ax-1 6 . . . . 5 ( 𝑥 ∈ {∅, 𝑂} → (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂}))
5958ralimi 3099 . . . 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 1158 . 2 (𝑂𝑉 → (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → 𝑥 ∈ {∅, 𝑂})))
62 issiga 30642 . . 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 578 1 (𝑂𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3731  cun 3732  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336  {cpr 4338   cuni 4596   class class class wbr 4811  cfv 6070  ωcom 7267  cdom 8162  sigAlgebracsiga 30638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-siga 30639
This theorem is referenced by: (None)
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