Theorem rintm 3913

Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
rintm	|- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )

Distinct variable group:   x, X

Allowed substitution hint:   A( x)

Proof of Theorem rintm

Step	Hyp	Ref	Expression
1		incom 3273	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2m 3803	. . . 4 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ U. ~P A )
3		ssid 3122	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 3905	. . . . 5 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
5	3, 4	mpbi 144	. . . 4 |- U. ~P A C_ A
6	2, 5	sstrdi 3114	. . 3 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ A )
7		df-ss 3089	. . 3 |- ( |^| X C_ A <-> ( |^| X i^i A ) = |^| X )
8	6, 7	sylib 121	. 2 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( |^| X i^i A ) = |^| X )
9	1, 8	syl5eq 2185	1 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 1481   i^i cin 3075   C_ wss 3076   ~Pcpw 3515   U.cuni 3744   |^|cint 3779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517  df-uni 3745  df-int 3780

This theorem is referenced by: (None)