Theorem intprg 3960

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3959. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Assertion

Ref	Expression
intprg	|- ((A e. V /\ B e. W) -> |^|{A, B} = (A i^i B))

Proof of Theorem intprg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 3799	. . . 4 |- (x = A -> {x, y} = {A, y})
2	1	inteqd 3931	. . 3 |- (x = A -> |^|{x, y} = |^|{A, y})
3		ineq1 3450	. . 3 |- (x = A -> (x i^i y) = (A i^i y))
4	2, 3	eqeq12d 2367	. 2 |- (x = A -> (|^|{x, y} = (x i^i y) <-> |^|{A, y} = (A i^i y)))
5		preq2 3800	. . . 4 |- (y = B -> {A, y} = {A, B})
6	5	inteqd 3931	. . 3 |- (y = B -> |^|{A, y} = |^|{A, B})
7		ineq2 3451	. . 3 |- (y = B -> (A i^i y) = (A i^i B))
8	6, 7	eqeq12d 2367	. 2 |- (y = B -> (|^|{A, y} = (A i^i y) <-> |^|{A, B} = (A i^i B)))
9		vex 2862	. . 3 |- x e. _V
10		vex 2862	. . 3 |- y e. _V
11	9, 10	intpr 3959	. 2 |- |^|{x, y} = (x i^i y)
12	4, 8, 11	vtocl2g 2918	1 |- ((A e. V /\ B e. W) -> |^|{A, B} = (A i^i B))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   i^i cin 3208  {cpr 3738  |^|cint 3926

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-sn 3741  df-pr 3742  df-int 3927

This theorem is referenced by:  intsng  3961