Theorem rintn0 5076

Description: Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)

Assertion

Ref	Expression
rintn0	⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Proof of Theorem rintn0

Step	Hyp	Ref	Expression
1		intssuni2 4940	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
2		ssid 3972	. . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
3		sspwuni 5067	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
4	2, 3	mpbi 230	. . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
5	1, 4	sstrdi 3962	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴)
6		sseqin2 4189	. 2 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)
7	5, 6	sylib 218	1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ≠ wne 2926   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  ∪ cuni 4874  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-nul 4300  df-pw 4568  df-uni 4875  df-int 4914

This theorem is referenced by:  mrerintcl  17565  ismred2  17571