Theorem intpr 3720

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 1415	. . . 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 2622	. . . . . . . 8 |- y e. _V
3	2	elpr 3467	. . . . . . 7 |- ( y e. { A , B } <-> ( y = A \/ y = B ) )
4	3	imbi1i 236	. . . . . 6 |- ( ( y e. { A , B } -> x e. y ) <-> ( ( y = A \/ y = B ) -> x e. y ) )
5		jaob 666	. . . . . 6 |- ( ( ( y = A \/ y = B ) -> x e. y ) <-> ( ( y = A -> x e. y ) /\ ( y = B -> x e. y ) ) )
6	4, 5	bitri 182	. . . . 5 |- ( ( y e. { A , B } -> x e. y ) <-> ( ( y = A -> x e. y ) /\ ( y = B -> x e. y ) ) )
7	6	albii 1404	. . . 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 2753	. . . . 5 |- ( x e. A <-> A. y ( y = A -> x e. y ) )
10		intpr.2	. . . . . 6 |- B e. _V
11	10	clel4 2753	. . . . 5 |- ( x e. B <-> A. y ( y = B -> x e. y ) )
12	9, 11	anbi12i 448	. . . 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 210	. . 3 |- ( A. y ( y e. { A , B } -> x e. y ) <-> ( x e. A /\ x e. B ) )
14		vex 2622	. . . 4 |- x e. _V
15	14	elint 3694	. . 3 |- ( x e. |^| { A , B } <-> A. y ( y e. { A , B } -> x e. y ) )
16		elin 3183	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
17	13, 15, 16	3bitr4i 210	. 2 |- ( x e. |^| { A , B } <-> x e. ( A i^i B ) )
18	17	eqriv 2085	1 |- |^| { A , B } = ( A i^i B )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 664   A.wal 1287   = wceq 1289   e. wcel 1438   _Vcvv 2619   i^i cin 2998   {cpr 3447   |^|cint 3688

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-sn 3452  df-pr 3453  df-int 3689

This theorem is referenced by:  intprg  3721  op1stb  4300  onintexmid  4388