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Theorem salgenn0 43573
Description: The set used in the definition of the generated sigma-algebra, is not empty. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Assertion
Ref Expression
salgenn0 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
Distinct variable group:   𝑋,𝑠
Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem salgenn0
StepHypRef Expression
1 uniexg 7546 . . . . 5 (𝑋𝑉 𝑋 ∈ V)
2 pwsal 43559 . . . . 5 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
31, 2syl 17 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
4 unipw 5349 . . . . . 6 𝒫 𝑋 = 𝑋
54a1i 11 . . . . 5 (𝑋𝑉 𝒫 𝑋 = 𝑋)
6 pwuni 4872 . . . . . 6 𝑋 ⊆ 𝒫 𝑋
76a1i 11 . . . . 5 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
85, 7jca 515 . . . 4 (𝑋𝑉 → ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋))
93, 8jca 515 . . 3 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
10 unieq 4844 . . . . . 6 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
1110eqeq1d 2740 . . . . 5 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
12 sseq2 3941 . . . . 5 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
1311, 12anbi12d 634 . . . 4 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
1413elrab 3614 . . 3 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
159, 14sylibr 237 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
1615ne0d 4264 1 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  wne 2941  {crab 3066  Vcvv 3420  wss 3880  c0 4251  𝒫 cpw 4527   cuni 4833  SAlgcsalg 43552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-ral 3067  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-pw 4529  df-sn 4556  df-pr 4558  df-uni 4834  df-salg 43553
This theorem is referenced by:  salgencl  43574  salgenuni  43579
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