Theorem bj-ismoored2 37074

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 4997	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴)
4	1, 2, 3	syl2anc 583	. . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴)
5		sseqin2 4244	. . 3 ⊢ (∩ 𝐵 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵)
6	4, 5	sylib 218	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵)
7		bj-ismoored.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Moore)
8	7, 1	bj-ismoored 37073	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)
9	6, 8	eqeltrrd 2845	1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ∩ cint 4970  Moorecmoore 37069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993  df-nul 4353  df-pw 4624  df-uni 4932  df-int 4971  df-bj-moore 37070

This theorem is referenced by: (None)