Theorem rintn0 5057

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 4923	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
2		ssid 3957	. . . 4 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
3		sspwuni 5048	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
4	2, 3	mpbi 230	. . 3 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
5	1, 4	sstrdi 3947	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑋 ⊆ 𝐴)
6		sseqin2 4173	. 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 1541   ≠ wne 2928   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550  ∪ cuni 4859  ∩ cint 4897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4284  df-pw 4552  df-uni 4860  df-int 4898

This theorem is referenced by:  mrerintcl  17496  ismred2  17502