Theorem bj-ismooredr 37419

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 4549	. . . 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 3129	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
6		bj-ismoore 37415	. 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 2114  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ∩ cint 4890  Moorecmoore 37413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-in 3897  df-ss 3907  df-nul 4275  df-pw 4544  df-uni 4852  df-int 4891  df-bj-moore 37414

This theorem is referenced by:  bj-ismooredr2  37420  bj-discrmoore  37421