Theorem bj-ismoored0 38218

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 38217	. 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
2		0elpw 5321	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
3		rint0 4951	. . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴)
4	3	eleq1d 2849	. . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
5	4	rspcv 3576	. . 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 220	1 ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910  Moorecmoore 38215

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-nul 4282  df-pw 4562  df-uni 4871  df-int 4911  df-bj-moore 38216

This theorem is used by:  bj-0nmoore  38224