Theorem bj-ismoorec 33060

Description: Characterization of Moore collections. (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-ismoorec	⊢ (𝐴 ∈ Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-ismoorec

Step	Hyp	Ref	Expression
1		elex 3212	. 2 ⊢ (𝐴 ∈ Moore → 𝐴 ∈ V)
2		bj-ismoore 33059	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
3	1, 2	biadan2 674	1 ⊢ (𝐴 ∈ Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475  Moorecmoore 33057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-bj-moore 33058

This theorem is referenced by:  bj-ismoored  33062