Theorem intpr 2617

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 1108	. . . 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 1860	. . . . . . . 8 |- y e. V
3	2	elpr 2476	. . . . . . 7 |- (y e. {A, B} <-> (y = A \/ y = B))
4	3	imbi1i 193	. . . . . 6 |- ((y e. {A, B} -> x e. y) <-> ((y = A \/ y = B) -> x e. y))
5		jaob 431	. . . . . 6 |- (((y = A \/ y = B) -> x e. y) <-> ((y = A -> x e. y) /\ (y = B -> x e. y)))
6	4, 5	bitri 180	. . . . 5 |- ((y e. {A, B} -> x e. y) <-> ((y = A -> x e. y) /\ (y = B -> x e. y)))
7	6	albii 1040	. . . 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 1941	. . . . 5 |- (x e. A <-> A.y(y = A -> x e. y))
10		intpr.2	. . . . . 6 |- B e. V
11	10	clel4 1941	. . . . 5 |- (x e. B <-> A.y(y = B -> x e. y))
12	9, 11	anbi12i 493	. . . 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 190	. . 3 |- (A.y(y e. {A, B} -> x e. y) <-> (x e. A /\ x e. B))
14		visset 1860	. . . 4 |- x e. V
15	14	elint 2593	. . 3 |- (x e. |^|{A, B} <-> A.y(y e. {A, B} -> x e. y))
16		elin 2258	. . 3 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
17	13, 15, 16	3bitr4i 190	. 2 |- (x e. |^|{A, B} <-> x e. (A i^i B))
18	17	eqriv 1519	1 |- |^|{A, B} = (A i^i B)

Syntax hints:   -> wi 3   \/ wo 229   /\ wa 230  A.wal 995   = wceq 997   e. wcel 999  Vcvv 1858   i^i cin 2097  {cpr 2462  |^|cint 2587

This theorem is referenced by:  intsn 2618  op1stb 2970  fiint 4620  shincli 9414  chincli 9466  intprd 10552

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-v 1859  df-un 2101  df-in 2102  df-sn 2464  df-pr 2465  df-int 2588

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