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Theorem pwsal 42477
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 5247 . . . 4 ∅ ∈ 𝒫 𝑋
21a1i 11 . . 3 (𝑋𝑉 → ∅ ∈ 𝒫 𝑋)
3 unipw 5333 . . . . . . . 8 𝒫 𝑋 = 𝑋
43difeq1i 4092 . . . . . . 7 ( 𝒫 𝑋𝑦) = (𝑋𝑦)
54a1i 11 . . . . . 6 (𝑋𝑉 → ( 𝒫 𝑋𝑦) = (𝑋𝑦))
6 difssd 4106 . . . . . . 7 (𝑋𝑉 → (𝑋𝑦) ⊆ 𝑋)
7 difexg 5222 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ V)
8 elpwg 4541 . . . . . . . 8 ((𝑋𝑦) ∈ V → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
97, 8syl 17 . . . . . . 7 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
106, 9mpbird 258 . . . . . 6 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
115, 10eqeltrd 2910 . . . . 5 (𝑋𝑉 → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1211adantr 481 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1312ralrimiva 3179 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
14 elpwi 4547 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
15 uniss 4851 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝑋 𝑦 𝒫 𝑋)
1614, 15syl 17 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 𝒫 𝑋)
1716, 3sseqtrdi 4014 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦𝑋)
18 vuniex 7455 . . . . . . . . 9 𝑦 ∈ V
1918a1i 11 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ V)
20 elpwg 4541 . . . . . . . 8 ( 𝑦 ∈ V → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2119, 20syl 17 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2217, 21mpbird 258 . . . . . 6 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ 𝒫 𝑋)
2322adantl 482 . . . . 5 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → 𝑦 ∈ 𝒫 𝑋)
2423a1d 25 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
2524ralrimiva 3179 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
262, 13, 253jca 1120 . 2 (𝑋𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋)))
27 pwexg 5270 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
28 issal 42476 . . 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 258 1 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  ωcom 7569  cdom 8495  SAlgcsalg 42470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-sn 4558  df-pr 4560  df-uni 4831  df-salg 42471
This theorem is referenced by:  salgenval  42483  salgenn0  42491  salgencntex  42503  psmeasure  42630
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