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Theorem pwsal 39839
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 4794 . . . 4 ∅ ∈ 𝒫 𝑋
21a1i 11 . . 3 (𝑋𝑉 → ∅ ∈ 𝒫 𝑋)
3 unipw 4879 . . . . . . . 8 𝒫 𝑋 = 𝑋
43difeq1i 3702 . . . . . . 7 ( 𝒫 𝑋𝑦) = (𝑋𝑦)
54a1i 11 . . . . . 6 (𝑋𝑉 → ( 𝒫 𝑋𝑦) = (𝑋𝑦))
6 difssd 3716 . . . . . . 7 (𝑋𝑉 → (𝑋𝑦) ⊆ 𝑋)
7 difexg 4768 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ V)
8 elpwg 4138 . . . . . . . 8 ((𝑋𝑦) ∈ V → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
97, 8syl 17 . . . . . . 7 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
106, 9mpbird 247 . . . . . 6 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
115, 10eqeltrd 2698 . . . . 5 (𝑋𝑉 → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1211adantr 481 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1312ralrimiva 2960 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
14 elpwi 4140 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
15 uniss 4424 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝑋 𝑦 𝒫 𝑋)
1614, 15syl 17 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 𝒫 𝑋)
1716, 3syl6sseq 3630 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦𝑋)
18 vuniex 6907 . . . . . . . . 9 𝑦 ∈ V
1918a1i 11 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ V)
20 elpwg 4138 . . . . . . . 8 ( 𝑦 ∈ V → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2119, 20syl 17 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 → ( 𝑦 ∈ 𝒫 𝑋 𝑦𝑋))
2217, 21mpbird 247 . . . . . 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 2960 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
262, 13, 253jca 1240 . 2 (𝑋𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋)))
27 pwexg 4810 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
28 issal 39838 . . 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 247 1 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  ωcom 7012  cdom 7897  SAlgcsalg 39832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151  df-uni 4403  df-salg 39833
This theorem is referenced by:  salgenval  39845  salgenn0  39853  salgencntex  39865  psmeasure  39992
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