Theorem rintm 3974

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 3325	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2m 3864	. . . 4 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ U. ~P A )
3		ssid 3173	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 3966	. . . . 5 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
5	3, 4	mpbi 145	. . . 4 |- U. ~P A C_ A
6	2, 5	sstrdi 3165	. . 3 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ A )
7		df-ss 3140	. . 3 |- ( |^| X C_ A <-> ( |^| X i^i A ) = |^| X )
8	6, 7	sylib 122	. 2 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( |^| X i^i A ) = |^| X )
9	1, 8	eqtrid 2220	1 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1490   e. wcel 2146   i^i cin 3126   C_ wss 3127   ~Pcpw 3572   U.cuni 3805   |^|cint 3840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574  df-uni 3806  df-int 3841

This theorem is referenced by: (None)