Theorem rintm 3765

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 3158	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2m 3660	. . . 4 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ U. ~P A )
3		ssid 3018	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 3760	. . . . 5 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
5	3, 4	mpbi 143	. . . 4 |- U. ~P A C_ A
6	2, 5	syl6ss 3011	. . 3 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ A )
7		df-ss 2986	. . 3 |- ( |^| X C_ A <-> ( |^| X i^i A ) = |^| X )
8	6, 7	sylib 120	. 2 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( |^| X i^i A ) = |^| X )
9	1, 8	syl5eq 2125	1 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   i^i cin 2972   C_ wss 2973   ~Pcpw 3382   U.cuni 3601   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-uni 3602  df-int 3637

This theorem is referenced by: (None)