Theorem intpr 3906

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 1495	. . . 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 2766	. . . . . . . 8 |- y e. _V
3	2	elpr 3643	. . . . . . 7 |- ( y e. { A , B } <-> ( y = A \/ y = B ) )
4	3	imbi1i 238	. . . . . 6 |- ( ( y e. { A , B } -> x e. y ) <-> ( ( y = A \/ y = B ) -> x e. y ) )
5		jaob 711	. . . . . 6 |- ( ( ( y = A \/ y = B ) -> x e. y ) <-> ( ( y = A -> x e. y ) /\ ( y = B -> x e. y ) ) )
6	4, 5	bitri 184	. . . . 5 |- ( ( y e. { A , B } -> x e. y ) <-> ( ( y = A -> x e. y ) /\ ( y = B -> x e. y ) ) )
7	6	albii 1484	. . . 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 2900	. . . . 5 |- ( x e. A <-> A. y ( y = A -> x e. y ) )
10		intpr.2	. . . . . 6 |- B e. _V
11	10	clel4 2900	. . . . 5 |- ( x e. B <-> A. y ( y = B -> x e. y ) )
12	9, 11	anbi12i 460	. . . 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 212	. . 3 |- ( A. y ( y e. { A , B } -> x e. y ) <-> ( x e. A /\ x e. B ) )
14		vex 2766	. . . 4 |- x e. _V
15	14	elint 3880	. . 3 |- ( x e. |^| { A , B } <-> A. y ( y e. { A , B } -> x e. y ) )
16		elin 3346	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
17	13, 15, 16	3bitr4i 212	. 2 |- ( x e. |^| { A , B } <-> x e. ( A i^i B ) )
18	17	eqriv 2193	1 |- |^| { A , B } = ( A i^i B )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   A.wal 1362   = wceq 1364   e. wcel 2167   _Vcvv 2763   i^i cin 3156   {cpr 3623   |^|cint 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-int 3875

This theorem is referenced by:  intprg  3907  op1stb  4513  onintexmid  4609