Theorem eliin 1999

Description: Membership in indexed intersection.

Assertion

Ref	Expression
eliin	|- (A e. D -> (A e. |^|x e. B C <-> A.x e. B A e. C))

Distinct variable group(s):   x,A

Proof of Theorem eliin

Step	Hyp	Ref	Expression
1		eleq1 1149	. . 3 |- (y = A -> (y e. C <-> A e. C))
2	1	biraldv 1219	. 2 |- (y = A -> (A.x e. B y e. C <-> A.x e. B A e. C))
3		df-iin 1997	. 2 |- |^|x e. B C = {y | A.x e. B y e. C}
4	2, 3	elab2g 1418	1 |- (A e. D -> (A e. |^|x e. B C <-> A.x e. B A e. C))

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092  A.wral 1201  |^|ciin 1995

This theorem is referenced by:  iuniin 2001  ssiin 2024  iinss 2025  iinun2 2031  iundif2 2032  iindif2 2033  iinuni 2036  iinpw 2038

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-iin 1997

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