Theorem bj-ismoored2 36510

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)
bj-ismoored2.3	⊢ (𝜑 → 𝐵 ≠ ∅)

Assertion

Ref	Expression
bj-ismoored2	⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)

Proof of Theorem bj-ismoored2

Step	Hyp	Ref	Expression
1		bj-ismoored.2	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
2		bj-ismoored2.3	. . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅)
3		intssuni2 4971	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴)
4	1, 2, 3	syl2anc 583	. . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴)
5		sseqin2 4211	. . 3 ⊢ (∩ 𝐵 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵)
6	4, 5	sylib 217	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵)
7		bj-ismoored.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Moore)
8	7, 1	bj-ismoored 36509	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)
9	6, 8	eqeltrrd 2829	1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   ≠ wne 2935   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  ∪ cuni 4903  ∩ cint 4944  Moorecmoore 36505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-pw 4600  df-uni 4904  df-int 4945  df-bj-moore 36506

This theorem is referenced by: (None)