Theorem elintg 2545

Description: Membership in class intersection, with the sethood requirement expressed as an antecedent.

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem elintg

Step	Hyp	Ref	Expression
1		eleq1 1537	. . 3 |- (y = A -> (y e. |^|B <-> A e. |^|B))
2		eleq1 1537	. . . 4 |- (y = A -> (y e. x <-> A e. x))
3	2	ralbidv 1666	. . 3 |- (y = A -> (A.x e. B y e. x <-> A.x e. B A e. x))
4	1, 3	bibi12d 631	. 2 |- (y = A -> ((y e. |^|B <-> A.x e. B y e. x) <-> (A e. |^|B <-> A.x e. B A e. x)))
5		visset 1816	. . 3 |- y e. V
6	5	elint2 2544	. 2 |- (y e. |^|B <-> A.x e. B y e. x)
7	4, 6	vtoclg 1850	1 |- (A e. C -> (A e. |^|B <-> A.x e. B A e. x))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  A.wral 1648  |^|cint 2537

This theorem is referenced by:  onmindif 3066  onmindif2 3067

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-int 2538

Copyright terms: Public domain