Theorem ssint 2549

Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse.

Assertion

Ref	Expression
ssint	|- (A (_ |^|B <-> A.x e. B A (_ x)

Distinct variable groups:   x,A   x,B

Proof of Theorem ssint

Step	Hyp	Ref	Expression
1		dfss3 2059	. 2 |- (A (_ |^|B <-> A.y e. A y e. |^|B)
2		visset 1813	. . . 4 |- y e. V
3	2	elint2 2540	. . 3 |- (y e. |^|B <-> A.x e. B y e. x)
4	3	ralbii 1667	. 2 |- (A.y e. A y e. |^|B <-> A.y e. A A.x e. B y e. x)
5		ralcom 1774	. . 3 |- (A.y e. A A.x e. B y e. x <-> A.x e. B A.y e. A y e. x)
6		dfss3 2059	. . . 4 |- (A (_ x <-> A.y e. A y e. x)
7	6	ralbii 1667	. . 3 |- (A.x e. B A (_ x <-> A.x e. B A.y e. A y e. x)
8	5, 7	bitr4 176	. 2 |- (A.y e. A A.x e. B y e. x <-> A.x e. B A (_ x)
9	1, 4, 8	3bitr 177	1 |- (A (_ |^|B <-> A.x e. B A (_ x)

Syntax hints:   <-> wb 146   e. wcel 958  A.wral 1645   (_ wss 2047  |^|cint 2533

This theorem is referenced by:  ssintab 2550  ssintub 2551  iinpw 2617  oneqmini 3017  fint 3650  iscard2 4854

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-int 2534

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