Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwsal Structured version   Visualization version   GIF version

Theorem pwsal 39211
 Description: The power set of a given set is a sigma-algebra (the so called discrete sigma-algebra). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
pwsal (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)

Proof of Theorem pwsal
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 0elpw 4760 . . . 4 ∅ ∈ 𝒫 𝑋
21a1i 11 . . 3 (𝑋𝑉 → ∅ ∈ 𝒫 𝑋)
3 unipw 4845 . . . . . . . 8 𝒫 𝑋 = 𝑋
43difeq1i 3686 . . . . . . 7 ( 𝒫 𝑋𝑦) = (𝑋𝑦)
54a1i 11 . . . . . 6 (𝑋𝑉 → ( 𝒫 𝑋𝑦) = (𝑋𝑦))
6 difssd 3700 . . . . . . 7 (𝑋𝑉 → (𝑋𝑦) ⊆ 𝑋)
7 difexg 4735 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ V)
8 elpwg 4116 . . . . . . . 8 ((𝑋𝑦) ∈ V → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
97, 8syl 17 . . . . . . 7 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
106, 9mpbird 246 . . . . . 6 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
115, 10eqeltrd 2688 . . . . 5 (𝑋𝑉 → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1211adantr 480 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1312ralrimiva 2949 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
14 elpwi 4117 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
15 uniss 4394 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝑋 𝑦 𝒫 𝑋)
1614, 15syl 17 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 𝒫 𝑋)
1716, 3syl6sseq 3614 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦𝑋)
18 vuniex 6852 . . . . . . . . 9 𝑦 ∈ V
1918a1i 11 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ V)
20 elpwg 4116 . . . . . . . 8 ( 𝑦 ∈ V → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2119, 20syl 17 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2217, 21mpbird 246 . . . . . 6 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ 𝒫 𝑋)
2322adantl 481 . . . . 5 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → 𝑦 ∈ 𝒫 𝑋)
2423a1d 25 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
2524ralrimiva 2949 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
262, 13, 253jca 1235 . 2 (𝑋𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋)))
27 pwexg 4776 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
28 issal 39210 . . 3 (𝒫 𝑋 ∈ V → (𝒫 𝑋 ∈ SAlg ↔ (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))))
2927, 28syl 17 . 2 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ↔ (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))))
3026, 29mpbird 246 1 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ωcom 6957   ≼ cdom 7839  SAlgcsalg 39204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-salg 39205 This theorem is referenced by:  salgenval  39217  salgenn0  39225  salgencntex  39237  psmeasure  39364
 Copyright terms: Public domain W3C validator