Theorem bj-ismooredr 36644

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 4605	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
2		bj-ismooredr.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
3	2	ex 411	. . . 4 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
4	1, 3	syl5 34	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
5	4	ralrimiv 3135	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
6		bj-ismoore 36640	. 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
7	5, 6	sylibr 233	1 ⊢ (𝜑 → 𝐴 ∈ Moore)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  ∀wral 3051   ∩ cin 3939   ⊆ wss 3940  𝒫 cpw 4598  ∪ cuni 4903  ∩ cint 4944  Moorecmoore 36638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-in 3947  df-ss 3957  df-nul 4319  df-pw 4600  df-uni 4904  df-int 4945  df-bj-moore 36639

This theorem is referenced by:  bj-ismooredr2  36645  bj-discrmoore  36646