Theorem bj-ismoored 37090

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 4954	. . . 4 ⊢ (𝑥 = 𝐵 → ∩ 𝑥 = ∩ 𝐵)
2	1	ineq2d 4228	. . 3 ⊢ (𝑥 = 𝐵 → (∪ 𝐴 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝐵))
3	2	eleq1d 2824	. 2 ⊢ (𝑥 = 𝐵 → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
4		bj-ismoored.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Moore)
5		bj-ismoore 37088	. . 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 5334	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴)
9	3, 6, 8	rspcdva 3623	1 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  ∩ cint 4951  Moorecmoore 37086

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-int 4952  df-bj-moore 37087

This theorem is referenced by:  bj-ismoored2  37091