Theorem rintn0 4619

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		intssuni2 4502	. . 3 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ U. ~P A )
2		ssid 3624	. . . 4 |- ~P A C_ ~P A
3		sspwuni 4611	. . . 4 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
4	2, 3	mpbi 220	. . 3 |- U. ~P A C_ A
5	1, 4	syl6ss 3615	. 2 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ A )
6		sseqin2 3817	. 2 |- ( |^| X C_ A <-> ( A i^i |^| X ) = |^| X )
7	5, 6	sylib 208	1 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-int 4476

This theorem is referenced by:  mrerintcl  16257  ismred2  16263