Theorem bj-ismoored2 37131

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 4954	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴)
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴)
5		sseqin2 4203	. . 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 37130	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)
9	6, 8	eqeltrrd 2836	1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2933   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  ∪ cuni 4888  ∩ cint 4927  Moorecmoore 37126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-in 3938  df-ss 3948  df-nul 4314  df-pw 4582  df-uni 4889  df-int 4928  df-bj-moore 37127

This theorem is referenced by: (None)