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Theorem pwsal 40955
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 4939 . . . 4 ∅ ∈ 𝒫 𝑋
21a1i 11 . . 3 (𝑋𝑉 → ∅ ∈ 𝒫 𝑋)
3 unipw 5023 . . . . . . . 8 𝒫 𝑋 = 𝑋
43difeq1i 3832 . . . . . . 7 ( 𝒫 𝑋𝑦) = (𝑋𝑦)
54a1i 11 . . . . . 6 (𝑋𝑉 → ( 𝒫 𝑋𝑦) = (𝑋𝑦))
6 difssd 3846 . . . . . . 7 (𝑋𝑉 → (𝑋𝑦) ⊆ 𝑋)
7 difexg 4916 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ V)
8 elpwg 4274 . . . . . . . 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 2803 . . . . 5 (𝑋𝑉 → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1211adantr 472 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → ( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
1312ralrimiva 3068 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋)
14 elpwi 4276 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
15 uniss 4566 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝑋 𝑦 𝒫 𝑋)
1614, 15syl 17 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 𝒫 𝑋)
1716, 3syl6sseq 3757 . . . . . . 7 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦𝑋)
18 vuniex 7071 . . . . . . . . 9 𝑦 ∈ V
1918a1i 11 . . . . . . . 8 (𝑦 ∈ 𝒫 𝒫 𝑋 𝑦 ∈ V)
20 elpwg 4274 . . . . . . . 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 473 . . . . 5 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → 𝑦 ∈ 𝒫 𝑋)
2423a1d 25 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
2524ralrimiva 3068 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋))
262, 13, 253jca 1379 . 2 (𝑋𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋( 𝒫 𝑋𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → 𝑦 ∈ 𝒫 𝑋)))
27 pwexg 4955 . . 3 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
28 issal 40954 . . 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 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  Vcvv 3304  cdif 3677  wss 3680  c0 4023  𝒫 cpw 4266   cuni 4544   class class class wbr 4760  ωcom 7182  cdom 8070  SAlgcsalg 40948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-pw 4268  df-sn 4286  df-pr 4288  df-uni 4545  df-salg 40949
This theorem is referenced by:  salgenval  40961  salgenn0  40969  salgencntex  40981  psmeasure  41108
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