Theorem intpr 3997

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 1534	. . . 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 2824	. . . . . . . 8 |- y e. _V
3	2	elpr 3726	. . . . . . 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 722	. . . . . 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 1523	. . . 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 2962	. . . . 5 |- ( x e. A <-> A. y ( y = A -> x e. y ) )
10		intpr.2	. . . . . 6 |- B e. _V
11	10	clel4 2962	. . . . 5 |- ( x e. B <-> A. y ( y = B -> x e. y ) )
12	9, 11	anbi12i 464	. . . 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 2824	. . . 4 |- x e. _V
15	14	elint 3971	. . 3 |- ( x e. |^| { A , B } <-> A. y ( y e. { A , B } -> x e. y ) )
16		elin 3412	. . 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 2235	1 |- |^| { A , B } = ( A i^i B )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   A.wal 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   {cpr 3706   |^|cint 3965

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-sn 3711  df-pr 3712  df-int 3966

This theorem is referenced by:  intprg  3998  op1stb  4619  onintexmid  4715