Theorem intpr 3839

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 1461	. . . 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 2715	. . . . . . . 8 |- y e. _V
3	2	elpr 3581	. . . . . . 7 |- ( y e. { A , B } <-> ( y = A \/ y = B ) )
4	3	imbi1i 237	. . . . . 6 |- ( ( y e. { A , B } -> x e. y ) <-> ( ( y = A \/ y = B ) -> x e. y ) )
5		jaob 700	. . . . . 6 |- ( ( ( y = A \/ y = B ) -> x e. y ) <-> ( ( y = A -> x e. y ) /\ ( y = B -> x e. y ) ) )
6	4, 5	bitri 183	. . . . 5 |- ( ( y e. { A , B } -> x e. y ) <-> ( ( y = A -> x e. y ) /\ ( y = B -> x e. y ) ) )
7	6	albii 1450	. . . 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 2848	. . . . 5 |- ( x e. A <-> A. y ( y = A -> x e. y ) )
10		intpr.2	. . . . . 6 |- B e. _V
11	10	clel4 2848	. . . . 5 |- ( x e. B <-> A. y ( y = B -> x e. y ) )
12	9, 11	anbi12i 456	. . . 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 211	. . 3 |- ( A. y ( y e. { A , B } -> x e. y ) <-> ( x e. A /\ x e. B ) )
14		vex 2715	. . . 4 |- x e. _V
15	14	elint 3813	. . 3 |- ( x e. |^| { A , B } <-> A. y ( y e. { A , B } -> x e. y ) )
16		elin 3290	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
17	13, 15, 16	3bitr4i 211	. 2 |- ( x e. |^| { A , B } <-> x e. ( A i^i B ) )
18	17	eqriv 2154	1 |- |^| { A , B } = ( A i^i B )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   A.wal 1333   = wceq 1335   e. wcel 2128   _Vcvv 2712   i^i cin 3101   {cpr 3561   |^|cint 3807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-sn 3566  df-pr 3567  df-int 3808

This theorem is referenced by:  intprg  3840  op1stb  4436  onintexmid  4530