Theorem bj-ismoored0 37094

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 37093	. 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
2		0elpw 5311	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
3		rint0 4952	. . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴)
4	3	eleq1d 2813	. . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
5	4	rspcv 3584	. . 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 217	1 ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3913  ∅c0 4296  𝒫 cpw 4563  ∪ cuni 4871  ∩ cint 4910  Moorecmoore 37091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4297  df-pw 4565  df-uni 4872  df-int 4911  df-bj-moore 37092

This theorem is referenced by:  bj-0nmoore  37100