Theorem bj-ismoored 37102

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

Hypotheses

Ref	Expression
bj-ismoored.1	⊢ (𝜑 → 𝐴 ∈ Moore)
bj-ismoored.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
bj-ismoored	⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)

Proof of Theorem bj-ismoored

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inteq 4916	. . . 4 ⊢ (𝑥 = 𝐵 → ∩ 𝑥 = ∩ 𝐵)
2	1	ineq2d 4186	. . 3 ⊢ (𝑥 = 𝐵 → (∪ 𝐴 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝐵))
3	2	eleq1d 2814	. 2 ⊢ (𝑥 = 𝐵 → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
4		bj-ismoored.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Moore)
5		bj-ismoore 37100	. . 3 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
6	4, 5	sylib 218	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
7		bj-ismoored.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
8	4, 7	sselpwd 5286	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴)
9	3, 6, 8	rspcdva 3592	1 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ∩ cint 4913  Moorecmoore 37098

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-nul 4300  df-pw 4568  df-uni 4875  df-int 4914  df-bj-moore 37099

This theorem is referenced by:  bj-ismoored2  37103