Theorem bj-ismoored0 34392

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 34391	. 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
2		0elpw 5248	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
3		rint0 4908	. . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴)
4	3	eleq1d 2897	. . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
5	4	rspcv 3617	. . 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 219	1 ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ∩ cin 3934  ∅c0 4290  𝒫 cpw 4538  ∪ cuni 4831  ∩ cint 4868  Moorecmoore 34389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-uni 4832  df-int 4869  df-bj-moore 34390

This theorem is referenced by:  bj-0nmoore  34398