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Theorem salgenn0 39225
 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 6853 . . . . 5 (𝑋𝑉 𝑋 ∈ V)
2 pwsal 39211 . . . . 5 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
31, 2syl 17 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
4 unipw 4845 . . . . . 6 𝒫 𝑋 = 𝑋
54a1i 11 . . . . 5 (𝑋𝑉 𝒫 𝑋 = 𝑋)
6 pwuni 4825 . . . . . 6 𝑋 ⊆ 𝒫 𝑋
76a1i 11 . . . . 5 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
85, 7jca 553 . . . 4 (𝑋𝑉 → ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋))
93, 8jca 553 . . 3 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
10 unieq 4380 . . . . . 6 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
1110eqeq1d 2612 . . . . 5 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
12 sseq2 3590 . . . . 5 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
1311, 12anbi12d 743 . . . 4 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
1413elrab 3331 . . 3 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
159, 14sylibr 223 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
16 ne0i 3880 . 2 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
1715, 16syl 17 1 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  SAlgcsalg 39204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-salg 39205 This theorem is referenced by:  salgencl  39226  salgenuni  39231
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