Theorem rintn0 5068

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 4932	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
2		ssid 3930	. . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
3		sspwuni 5059	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
4	2, 3	mpbi 233	. . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
5	1, 4	sstrdi 3920	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴)
6		sseqin2 4120	. 2 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)
7	5, 6	sylib 221	1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ≠ wne 2955   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-nul 4279  df-pw 4558  df-uni 4867  df-int 4907

This theorem is used by:  mrerintcl  17772  ismred2  17778