Theorem intpr 2558

Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42.

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

Step	Hyp	Ref	Expression
1		19.26 1065	. . . 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		visset 1809	. . . . . . . 8 |- y e. V
3	2	elpr 2420	. . . . . . 7 |- (y e. {A, B} <-> (y = A \/ y = B))
4	3	imbi1i 186	. . . . . 6 |- ((y e. {A, B} -> x e. y) <-> ((y = A \/ y = B) -> x e. y))
5		jaob 422	. . . . . 6 |- (((y = A \/ y = B) -> x e. y) <-> ((y = A -> x e. y) /\ (y = B -> x e. y)))
6	4, 5	bitr 173	. . . . 5 |- ((y e. {A, B} -> x e. y) <-> ((y = A -> x e. y) /\ (y = B -> x e. y)))
7	6	albii 997	. . . 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 1890	. . . . 5 |- (x e. A <-> A.y(y = A -> x e. y))
10		intpr.2	. . . . . 6 |- B e. V
11	10	clel4 1890	. . . . 5 |- (x e. B <-> A.y(y = B -> x e. y))
12	9, 11	anbi12i 482	. . . 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	3bitr4 183	. . 3 |- (A.y(y e. {A, B} -> x e. y) <-> (x e. A /\ x e. B))
14		visset 1809	. . . 4 |- x e. V
15	14	elint 2534	. . 3 |- (x e. |^|{A, B} <-> A.y(y e. {A, B} -> x e. y))
16		elin 2203	. . 3 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
17	13, 15, 16	3bitr4 183	. 2 |- (x e. |^|{A, B} <-> x e. (A i^i B))
18	17	eqriv 1472	1 |- |^|{A, B} = (A i^i B)

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  Vcvv 1807   i^i cin 2042  {cpr 2406  |^|cint 2528

This theorem is referenced by:  intsn 2559  op1stb 2908  fiint 4540  shincl 9269  chincl 9321  intprd 10403

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-in 2047  df-sn 2408  df-pr 2409  df-int 2529

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