Theorem bj-ismoored2 38220

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 4936	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ⊆ ∪ 𝐴)
4	1, 2, 3	syl2anc 596	. . 3 ⊢ (𝜑 → ∩ 𝐵 ⊆ ∪ 𝐴)
5		sseqin2 4123	. . 3 ⊢ (∩ 𝐵 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵)
6	4, 5	sylib 221	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) = ∩ 𝐵)
7		bj-ismoored.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Moore)
8	7, 1	bj-ismoored 38219	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)
9	6, 8	eqeltrrd 2865	1 ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  ∪ cuni 4870  ∩ cint 4910  Moorecmoore 38215

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-nul 4282  df-pw 4562  df-uni 4871  df-int 4911  df-bj-moore 38216

This theorem is used by: (None)