Theorem rintn0 4422

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 3696	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2 4313	. . . 4 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ U. ~P A )
3		ssid 3528	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 4417	. . . . 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 3521	. . 3 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ A )
7		df-ss 3495	. . 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 2520	1 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   =/= wne 2662   i^i cin 3480   C_ wss 3481   (/)c0 3790   ~Pcpw 4016   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 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-v 3120  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-uni 4252  df-int 4289

This theorem is referenced by:  mrerintcl  14869  ismred2  14875