Theorem rintn0 4057

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 ⊆ ℘A ∧ X ≠ ∅) → (A ∩ ∩X) = ∩X)

Proof of Theorem rintn0

Step	Hyp	Ref	Expression
1		incom 3449	. 2 ⊢ (A ∩ ∩X) = (∩X ∩ A)
2		intssuni2 3952	. . . 4 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → ∩X ⊆ ∪℘A)
3		ssid 3291	. . . . 5 ⊢ ℘A ⊆ ℘A
4		sspwuni 4052	. . . . 5 ⊢ (℘A ⊆ ℘A ↔ ∪℘A ⊆ A)
5	3, 4	mpbi 199	. . . 4 ⊢ ∪℘A ⊆ A
6	2, 5	syl6ss 3285	. . 3 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → ∩X ⊆ A)
7		df-ss 3260	. . 3 ⊢ (∩X ⊆ A ↔ (∩X ∩ A) = ∩X)
8	6, 7	sylib 188	. 2 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → (∩X ∩ A) = ∩X)
9	1, 8	syl5eq 2397	1 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → (A ∩ ∩X) = ∩X)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2517   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551  ℘cpw 3723  ∪cuni 3892  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-uni 3893  df-int 3928

This theorem is referenced by: (None)