Theorem rintm 3744

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 3129	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2m 3639	. . . 4 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ U. ~P A )
3		ssid 2964	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 3739	. . . . 5 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
5	3, 4	mpbi 133	. . . 4 |- U. ~P A C_ A
6	2, 5	syl6ss 2957	. . 3 |- ( ( X C_ ~P A /\ E. x x e. X ) -> |^| X C_ A )
7		df-ss 2931	. . 3 |- ( |^| X C_ A <-> ( |^| X i^i A ) = |^| X )
8	6, 7	sylib 127	. 2 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( |^| X i^i A ) = |^| X )
9	1, 8	syl5eq 2084	1 |- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   i^i cin 2916   C_ wss 2917   ~Pcpw 3359   U.cuni 3580   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581  df-int 3616

This theorem is referenced by: (None)