Theorem bj-discrmoore 37094

Description: The powerclass 𝒫 𝐴 is a Moore collection if and only if 𝐴 is a set. It is then called the discrete Moore collection. (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-discrmoore	⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)

Proof of Theorem bj-discrmoore

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unipw 5461	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	ineq1i 4224	. . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥)
3		inex1g 5325	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V)
4		inss1 4245	. . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴)
6	3, 5	elpwd 4611	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
7	2, 6	eqeltrid 2843	. . . 4 ⊢ (𝐴 ∈ V → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
8	7	adantr 480	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
9	8	bj-ismooredr 37092	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore)
10		pwexr 7784	. 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V)
11	9, 10	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)

Syntax hints:   ↔ wb 206   ∈ wcel 2106  Vcvv 3478   ∩ 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  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-int 4952  df-bj-moore 37087

This theorem is referenced by: (None)