Theorem bj-ismoored2 35206

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 4901	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴)
4	1, 2, 3	syl2anc 583	. . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴)
5		sseqin2 4146	. . 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 35205	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)
9	6, 8	eqeltrrd 2840	1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ∩ cint 4876  Moorecmoore 35201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-uni 4837  df-int 4877  df-bj-moore 35202

This theorem is referenced by: (None)