Theorem intpr 3668

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 1410	. . . 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 2604	. . . . . . . 8 |- y e. _V
3	2	elpr 3419	. . . . . . 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 663	. . . . . 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 1399	. . . 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 2731	. . . . 5 |- ( x e. A <-> A. y ( y = A -> x e. y ) )
10		intpr.2	. . . . . 6 |- B e. _V
11	10	clel4 2731	. . . . 5 |- ( x e. B <-> A. y ( y = B -> x e. y ) )
12	9, 11	anbi12i 447	. . . 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 2604	. . . 4 |- x e. _V
15	14	elint 3642	. . 3 |- ( x e. |^| { A , B } <-> A. y ( y e. { A , B } -> x e. y ) )
16		elin 3155	. . 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 2078	1 |- |^| { A , B } = ( A i^i B )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   _Vcvv 2601   i^i cin 2972   {cpr 3399   |^|cint 3636

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-sn 3404  df-pr 3405  df-int 3637

This theorem is referenced by:  intprg  3669  op1stb  4227  onintexmid  4315