Theorem bj-ismooredr 33949

Description: Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)

Hypotheses

Ref	Expression
bj-ismooredr.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
bj-ismooredr.2	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-ismooredr

Step	Hyp	Ref	Expression
1		elpwi 4426	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
2		bj-ismooredr.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
3	2	ex 405	. . . 4 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
4	1, 3	syl5 34	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
5	4	ralrimiv 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
6		bj-ismooredr.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		bj-ismoore 33944	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
9	5, 8	mpbird 249	1 ⊢ (𝜑 → 𝐴 ∈ Moore)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∈ wcel 2051  ∀wral 3081   ∩ cin 3821   ⊆ wss 3822  𝒫 cpw 4416  ∪ cuni 4708  ∩ cint 4745  Moorecmoore 33942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-in 3829  df-ss 3836  df-pw 4418  df-uni 4709  df-bj-moore 33943

This theorem is referenced by:  bj-discrmoore  33951