Theorem bj-discrmoore 34402

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 5342	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	ineq1i 4184	. . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥)
3		inex1g 5222	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V)
4		inss1 4204	. . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴)
6	3, 5	elpwd 4546	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
7	2, 6	eqeltrid 2917	. . . 4 ⊢ (𝐴 ∈ V → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
8	7	adantr 483	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
9	8	bj-ismooredr 34400	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore)
10		pwexr 7486	. 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V)
11	9, 10	impbii 211	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)

Syntax hints:   ↔ wb 208   ∈ wcel 2110  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  ∪ cuni 4837  ∩ cint 4875  Moorecmoore 34394

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4567  df-pr 4569  df-uni 4838  df-int 4876  df-bj-moore 34395

This theorem is referenced by: (None)