Theorem bj-ismoored0 35204

Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-ismoored0	⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)

Proof of Theorem bj-ismoored0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-ismoore 35203	. 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
2		0elpw 5273	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
3		rint0 4918	. . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴)
4	3	eleq1d 2823	. . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
5	4	rspcv 3547	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴))
6	2, 5	ax-mp 5	. 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴)
7	1, 6	sylbi 216	1 ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∩ cin 3882  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ∩ cint 4876  Moorecmoore 35201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-uni 4837  df-int 4877  df-bj-moore 35202

This theorem is referenced by:  bj-0nmoore  35210