Theorem bj-discrmoore 35209

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 5360	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	ineq1i 4139	. . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥)
3		inex1g 5238	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V)
4		inss1 4159	. . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴)
6	3, 5	elpwd 4538	. . . . 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 35207	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore)
10		pwexr 7593	. 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V)
11	9, 10	impbii 208	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)

Syntax hints:   ↔ wb 205   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ∩ cint 4876  Moorecmoore 35201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-pr 4561  df-uni 4837  df-int 4877  df-bj-moore 35202

This theorem is referenced by: (None)