Theorem intpr 3960

Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Hypotheses

Ref	Expression
intpr.1	|- A e. _V
intpr.2	|- B e. _V

Assertion

Ref	Expression
intpr	|- |^|{A, B} = (A i^i B)

Proof of Theorem intpr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.26 1593	. . . 4 |- (A.y((y = A -> x e. y) /\ (y = B -> x e. y)) <-> (A.y(y = A -> x e. y) /\ A.y(y = B -> x e. y)))
2		vex 2863	. . . . . . . 8 |- y e. _V
3	2	elpr 3752	. . . . . . 7 |- (y e. {A, B} <-> (y = A \/ y = B))
4	3	imbi1i 315	. . . . . 6 |- ((y e. {A, B} -> x e. y) <-> ((y = A \/ y = B) -> x e. y))
5		jaob 758	. . . . . 6 |- (((y = A \/ y = B) -> x e. y) <-> ((y = A -> x e. y) /\ (y = B -> x e. y)))
6	4, 5	bitri 240	. . . . 5 |- ((y e. {A, B} -> x e. y) <-> ((y = A -> x e. y) /\ (y = B -> x e. y)))
7	6	albii 1566	. . . 4 |- (A.y(y e. {A, B} -> x e. y) <-> A.y((y = A -> x e. y) /\ (y = B -> x e. y)))
8		intpr.1	. . . . . 6 |- A e. _V
9	8	clel4 2979	. . . . 5 |- (x e. A <-> A.y(y = A -> x e. y))
10		intpr.2	. . . . . 6 |- B e. _V
11	10	clel4 2979	. . . . 5 |- (x e. B <-> A.y(y = B -> x e. y))
12	9, 11	anbi12i 678	. . . 4 |- ((x e. A /\ x e. B) <-> (A.y(y = A -> x e. y) /\ A.y(y = B -> x e. y)))
13	1, 7, 12	3bitr4i 268	. . 3 |- (A.y(y e. {A, B} -> x e. y) <-> (x e. A /\ x e. B))
14		vex 2863	. . . 4 |- x e. _V
15	14	elint 3933	. . 3 |- (x e. |^|{A, B} <-> A.y(y e. {A, B} -> x e. y))
16		elin 3220	. . 3 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
17	13, 15, 16	3bitr4i 268	. 2 |- (x e. |^|{A, B} <-> x e. (A i^i B))
18	17	eqriv 2350	1 |- |^|{A, B} = (A i^i B)

Syntax hints:   -> wi 4   \/ wo 357   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  _Vcvv 2860   i^i cin 3209  {cpr 3739  |^|cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-sn 3742  df-pr 3743  df-int 3928

This theorem is referenced by:  intprg  3961  uniintsn  3964