Theorem bj-ismoored 37120

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 4898	. . . 4 ⊢ (𝑥 = 𝐵 → ∩ 𝑥 = ∩ 𝐵)
2	1	ineq2d 4168	. . 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 37118	. . 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 5264	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴)
9	3, 6, 8	rspcdva 3576	1 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2110  ∀wral 3045   ∩ cin 3899   ⊆ wss 3900  𝒫 cpw 4548  ∪ cuni 4857  ∩ cint 4895  Moorecmoore 37116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-in 3907  df-ss 3917  df-nul 4282  df-pw 4550  df-uni 4858  df-int 4896  df-bj-moore 37117

This theorem is referenced by:  bj-ismoored2  37121