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Theorem salgenn0 46187
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 7771 . . . . 5 (𝑋𝑉 𝑋 ∈ V)
2 pwsal 46171 . . . . 5 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
31, 2syl 17 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
4 unipw 5473 . . . . . 6 𝒫 𝑋 = 𝑋
54a1i 11 . . . . 5 (𝑋𝑉 𝒫 𝑋 = 𝑋)
6 pwuni 4971 . . . . . 6 𝑋 ⊆ 𝒫 𝑋
76a1i 11 . . . . 5 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
85, 7jca 511 . . . 4 (𝑋𝑉 → ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋))
93, 8jca 511 . . 3 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
10 unieq 4942 . . . . . 6 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
1110eqeq1d 2736 . . . . 5 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
12 sseq2 4029 . . . . 5 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
1311, 12anbi12d 631 . . . 4 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
1413elrab 3703 . . 3 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
159, 14sylibr 234 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
1615ne0d 4360 1 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2103  wne 2942  {crab 3438  Vcvv 3482  wss 3970  c0 4347  𝒫 cpw 4622   cuni 4931  SAlgcsalg 46164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932  df-salg 46165
This theorem is referenced by:  salgencl  46188  salgenuni  46193
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