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Theorem salgenn0 39853
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 6908 . . . . 5 (𝑋𝑉 𝑋 ∈ V)
2 pwsal 39839 . . . . 5 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
31, 2syl 17 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
4 unipw 4879 . . . . . 6 𝒫 𝑋 = 𝑋
54a1i 11 . . . . 5 (𝑋𝑉 𝒫 𝑋 = 𝑋)
6 pwuni 4859 . . . . . 6 𝑋 ⊆ 𝒫 𝑋
76a1i 11 . . . . 5 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
85, 7jca 554 . . . 4 (𝑋𝑉 → ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋))
93, 8jca 554 . . 3 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
10 unieq 4410 . . . . . 6 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
1110eqeq1d 2623 . . . . 5 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
12 sseq2 3606 . . . . 5 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
1311, 12anbi12d 746 . . . 4 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
1413elrab 3346 . . 3 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
159, 14sylibr 224 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
16 ne0i 3897 . 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 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402  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-ne 2791  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:  salgencl  39854  salgenuni  39859
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