Theorem bj-discrmoore 34526

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 5308	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	ineq1i 4135	. . . . 5 ⊢ (∪ 𝒫 𝐴 ∩ ∩ 𝑥) = (𝐴 ∩ ∩ 𝑥)
3		inex1g 5187	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ V)
4		inss1 4155	. . . . . . 7 ⊢ (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴
5	4	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ⊆ 𝐴)
6	3, 5	elpwd 4505	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
7	2, 6	eqeltrid 2894	. . . 4 ⊢ (𝐴 ∈ V → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
8	7	adantr 484	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → (∪ 𝒫 𝐴 ∩ ∩ 𝑥) ∈ 𝒫 𝐴)
9	8	bj-ismooredr 34524	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Moore)
10		pwexr 7467	. 2 ⊢ (𝒫 𝐴 ∈ Moore → 𝐴 ∈ V)
11	9, 10	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)

Syntax hints:   ↔ wb 209   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  ∩ cint 4838  Moorecmoore 34518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-pw 4499  df-sn 4526  df-pr 4528  df-uni 4801  df-int 4839  df-bj-moore 34519

This theorem is referenced by: (None)