Theorem rintn0 5045

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 4910	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
2		ssid 3944	. . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
3		sspwuni 5036	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
4	2, 3	mpbi 231	. . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
5	1, 4	sstrdi 3934	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴)
6		sseqin2 4159	. 2 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)
7	5, 6	sylib 219	1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ≠ wne 2935   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  ∪ cuni 4845  ∩ cint 4884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-in 3897  df-ss 3907  df-nul 4269  df-pw 4538  df-uni 4846  df-int 4885

This theorem is referenced by:  mrerintcl  17557  ismred2  17563