Theorem bj-ismoored 37462

Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)

Hypotheses

Ref	Expression
bj-ismoored.1	⊢ (𝜑 → 𝐴 ∈ Moore)
bj-ismoored.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
bj-ismoored	⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)

Proof of Theorem bj-ismoored

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inteq 4883	. . . 4 ⊢ (𝑥 = 𝐵 → ∩ 𝑥 = ∩ 𝐵)
2	1	ineq2d 4152	. . 3 ⊢ (𝑥 = 𝐵 → (∪ 𝐴 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝐵))
3	2	eleq1d 2821	. 2 ⊢ (𝑥 = 𝐵 → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴))
4		bj-ismoored.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Moore)
5		bj-ismoore 37460	. . 3 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
6	4, 5	sylib 219	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
7		bj-ismoored.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
8	4, 7	sselpwd 5259	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴)
9	3, 6, 8	rspcdva 3564	1 ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2115  ∀wral 3050   ∩ cin 3885   ⊆ wss 3886  𝒫 cpw 4532  ∪ cuni 4841  ∩ cint 4880  Moorecmoore 37458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-ext 2708  ax-sep 5221  ax-nul 5231  ax-pow 5297

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3051  df-rex 3061  df-rab 3389  df-v 3430  df-dif 3889  df-in 3893  df-ss 3903  df-nul 4265  df-pw 4534  df-uni 4842  df-int 4881  df-bj-moore 37459

This theorem is referenced by:  bj-ismoored2  37463