Theorem rintn0 4426

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 C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X )

Proof of Theorem rintn0

Step	Hyp	Ref	Expression
1		incom 3687	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2 4314	. . . 4 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ U. ~P A )
3		ssid 3518	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 4421	. . . . 5 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
5	3, 4	mpbi 208	. . . 4 |- U. ~P A C_ A
6	2, 5	syl6ss 3511	. . 3 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ A )
7		df-ss 3485	. . 3 |- ( |^| X C_ A <-> ( |^| X i^i A ) = |^| X )
8	6, 7	sylib 196	. 2 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( |^| X i^i A ) = |^| X )
9	1, 8	syl5eq 2510	1 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   =/= wne 2652   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U.cuni 4251   |^|cint 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794  df-pw 4017  df-uni 4252  df-int 4289

This theorem is referenced by:  mrerintcl  15014  ismred2  15020