Theorem bj-ismoored 33555

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		bj-ismoored.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
2		bj-ismoored.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Moore)
3		bj-ismoorec 33553	. . 3 ⊢ (𝐴 ∈ Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
4	2, 3	sylib 210	. 2 ⊢ (𝜑 → (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
5		elpw2g 5019	. . . . 5 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
6	5	biimparc 472	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ V) → 𝐵 ∈ 𝒫 𝐴)
7		inteq 4670	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ∩ 𝑥 = ∩ 𝐵)
8	7	ineq2d 4012	. . . . . 6 ⊢ (𝑥 = 𝐵 → (∪ 𝐴 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝐵))
9	8	eleq1d 2863	. . . . 5 ⊢ (𝑥 = 𝐵 → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
10	9	rspcv 3493	. . . 4 ⊢ (𝐵 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
11	6, 10	syl 17	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ V) → (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
12	11	expimpd 446	. 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
13	1, 4, 12	sylc 65	1 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3089  Vcvv 3385   ∩ cin 3768   ⊆ wss 3769  𝒫 cpw 4349  ∪ cuni 4628  ∩ cint 4667  Moorecmoore 33550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777  ax-sep 4975

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776  df-ss 3783  df-pw 4351  df-uni 4629  df-int 4668  df-bj-moore 33551

This theorem is referenced by:  bj-ismoored2  33556