Theorem bj-ismooredr 37092

Description: Sufficient condition to be a Moore collection. Note that there is no sethood hypothesis on 𝐴: it is a consequence of the only hypothesis. (Contributed by BJ, 9-Dec-2021.)

Hypothesis

Ref	Expression
bj-ismooredr.1	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)

Assertion

Ref	Expression
bj-ismooredr	⊢ (𝜑 → 𝐴 ∈ Moore)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

Proof of Theorem bj-ismooredr

Step	Hyp	Ref	Expression
1		elpwi 4612	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
2		bj-ismooredr.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
3	2	ex 412	. . . 4 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
4	1, 3	syl5 34	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
5	4	ralrimiv 3143	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
6		bj-ismoore 37088	. 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
7	5, 6	sylibr 234	1 ⊢ (𝜑 → 𝐴 ∈ Moore)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  ∀wral 3059   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  ∩ cint 4951  Moorecmoore 37086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-int 4952  df-bj-moore 37087

This theorem is referenced by:  bj-ismooredr2  37093  bj-discrmoore  37094