Theorem salgenn0 46786

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	⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)

Distinct variable group:   𝑋,𝑠

Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem salgenn0

Step	Hyp	Ref	Expression
1		uniexg 7686	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
2		pwsal 46770	. . . . 5 ⊢ (∪ 𝑋 ∈ V → 𝒫 ∪ 𝑋 ∈ SAlg)
3	1, 2	syl 17	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ SAlg)
4		unipw 5391	. . . . . 6 ⊢ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋
5	4	a1i 11	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋)
6		pwuni 4878	. . . . . 6 ⊢ 𝑋 ⊆ 𝒫 ∪ 𝑋
7	6	a1i 11	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝒫 ∪ 𝑋)
8	5, 7	jca 517	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋))
9	3, 8	jca 517	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝒫 ∪ 𝑋 ∈ SAlg ∧ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
10		unieq 4851	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ∪ 𝑠 = ∪ 𝒫 ∪ 𝑋)
11	10	eqeq1d 2743	. . . . 5 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋))
12		sseq2 3942	. . . . 5 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝒫 ∪ 𝑋))
13	11, 12	anbi12d 639	. . . 4 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
14	13	elrab 3630	. . 3 ⊢ (𝒫 ∪ 𝑋 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝒫 ∪ 𝑋 ∈ SAlg ∧ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
15	9, 14	sylibr 236	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
16	15	ne0d 4272	1 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  {crab 3393  Vcvv 3433   ⊆ wss 3884  ∅c0 4263  𝒫 cpw 4531  ∪ cuni 4840  SAlgcsalg 46763

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-pw 4533  df-sn 4558  df-pr 4560  df-uni 4841  df-salg 46764

This theorem is referenced by:  salgencl  46787  salgenuni  46792