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Theorem salgenn0 42813
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 7449 . . . . 5 (𝑋𝑉 𝑋 ∈ V)
2 pwsal 42799 . . . . 5 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
31, 2syl 17 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
4 unipw 5324 . . . . . 6 𝒫 𝑋 = 𝑋
54a1i 11 . . . . 5 (𝑋𝑉 𝒫 𝑋 = 𝑋)
6 pwuni 4856 . . . . . 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 4830 . . . . . 6 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
1110eqeq1d 2826 . . . . 5 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
12 sseq2 3977 . . . . 5 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
1311, 12anbi12d 633 . . . 4 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
1413elrab 3665 . . 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 4282 1 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3013  {crab 3136  Vcvv 3479  wss 3918  c0 4274  𝒫 cpw 4520   cuni 4819  SAlgcsalg 42792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-pw 4522  df-sn 4549  df-pr 4551  df-uni 4820  df-salg 42793
This theorem is referenced by:  salgencl  42814  salgenuni  42819
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