Theorem bj-discrmoore 37155

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 5389	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	ineq1i 4163	. . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥)
3		inex1g 5255	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V)
4		inss1 4184	. . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴)
6	3, 5	elpwd 4553	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
7	2, 6	eqeltrid 2835	. . . 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 37153	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore)
10		pwexr 7698	. 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V)
11	9, 10	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)

Syntax hints:   ↔ wb 206   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856  ∩ cint 4895  Moorecmoore 37147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-pw 4549  df-sn 4574  df-pr 4576  df-uni 4857  df-int 4896  df-bj-moore 37148

This theorem is referenced by: (None)