Theorem rintm 4100

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 3421	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2m 3989	. . . 4 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ U. ~P A )
3		ssid 3268	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 4092	. . . . 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 3260	. . 3 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ A )
7		df-ss 3233	. . 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 2283	1 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   i^i cin 3219   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   |^|cint 3965

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-uni 3931  df-int 3966

This theorem is referenced by: (None)