Theorem bj-ismoored 33062

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 33060	. . 3 ⊢ (𝐴 ∈ Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
4	2, 3	sylib 208	. 2 ⊢ (𝜑 → (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
5		elpw2g 4827	. . . . 5 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
6	5	biimparc 504	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ V) → 𝐵 ∈ 𝒫 𝐴)
7		inteq 4478	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ∩ 𝑥 = ∩ 𝐵)
8	7	ineq2d 3814	. . . . . 6 ⊢ (𝑥 = 𝐵 → (∪ 𝐴 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝐵))
9	8	eleq1d 2686	. . . . 5 ⊢ (𝑥 = 𝐵 → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
10	9	rspcv 3305	. . . 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 629	. 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 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475  Moorecmoore 33057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-int 4476  df-bj-moore 33058

This theorem is referenced by:  bj-ismoored2  33063