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 4939	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
2		ssid 3960	. . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
3		sspwuni 5067	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
4	2, 3	mpbi 233	. . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
5	1, 4	sstrdi 3950	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴)
6		sseqin2 4177	. 2 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)
7	5, 6	sylib 221	1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ≠ wne 2958   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  ∪ cuni 4873  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4288  df-pw 4565  df-uni 4874  df-int 4914

This theorem is referenced by:  mrerintcl  17650  ismred2  17656