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Theorem pwsal 45031
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 5355 . . . 4 ∅ ∈ 𝒫 𝑋
21a1i 11 . . 3 (𝑋𝑉 → ∅ ∈ 𝒫 𝑋)
3 unipw 5451 . . . . . . . 8 𝒫 𝑋 = 𝑋
43difeq1i 4119 . . . . . . 7 ( 𝒫 𝑋𝑦) = (𝑋𝑦)
54a1i 11 . . . . . 6 (𝑋𝑉 → ( 𝒫 𝑋𝑦) = (𝑋𝑦))
6 difssd 4133 . . . . . . 7 (𝑋𝑉 → (𝑋𝑦) ⊆ 𝑋)
7 difexg 5328 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ V)
8 elpwg 4606 . . . . . . . 8 ((𝑋𝑦) ∈ V → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
97, 8syl 17 . . . . . . 7 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
106, 9mpbird 257 . . . . . 6 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
115, 10eqeltrd 2834 . . . . 5 (𝑋𝑉 → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1211adantr 482 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1312ralrimiva 3147 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
14 elpwi 4610 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
1514unissd 4919 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 𝒫 𝑋)
1615, 3sseqtrdi 4033 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦𝑋)
17 vuniex 7729 . . . . . . . . 9 𝑦 ∈ V
1817a1i 11 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ V)
19 elpwg 4606 . . . . . . . 8 ( 𝑦 ∈ V → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2018, 19syl 17 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2116, 20mpbird 257 . . . . . 6 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ 𝒫 𝑋)
2221adantl 483 . . . . 5 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → 𝑦 ∈ 𝒫 𝑋)
2322a1d 25 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
2423ralrimiva 3147 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
252, 13, 243jca 1129 . 2 (𝑋𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋)))
26 pwexg 5377 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
27 issal 45030 . . 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))))
2826, 27syl 17 . 2 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ↔ (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))))
2925, 28mpbird 257 1 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cdif 3946  wss 3949  c0 4323  𝒫 cpw 4603   cuni 4909   class class class wbr 5149  ωcom 7855  cdom 8937  SAlgcsalg 45024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-salg 45025
This theorem is referenced by:  salgenval  45037  salgenn0  45047  salgencntex  45059  psmeasure  45187
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