Theorem unpr 1930

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.

Hypotheses

Ref	Expression
unpr.1	|- A e. V
unpr.2	|- B e. V

Assertion

Ref	Expression
unpr	|- U.{A, B} = (A u. B)

Proof of Theorem unpr

Step	Hyp	Ref	Expression
1		19.43 767	. . . 4 |- (E.y((x e. y /\ y = A) \/ (x e. y /\ y = B)) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
2		visset 1350	. . . . . . . 8 |- y e. V
3	2	elpr 1823	. . . . . . 7 |- (y e. {A, B} <-> (y = A \/ y = B))
4	3	anbi2i 367	. . . . . 6 |- ((x e. y /\ y e. {A, B}) <-> (x e. y /\ (y = A \/ y = B)))
5		andi 456	. . . . . 6 |- ((x e. y /\ (y = A \/ y = B)) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
6	4, 5	bitr 151	. . . . 5 |- ((x e. y /\ y e. {A, B}) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
7	6	biex 733	. . . 4 |- (E.y(x e. y /\ y e. {A, B}) <-> E.y((x e. y /\ y = A) \/ (x e. y /\ y = B)))
8		unpr.1	. . . . . . 7 |- A e. V
9	8	clel3 1375	. . . . . 6 |- (x e. A <-> E.y(y = A /\ x e. y))
10		exancom 736	. . . . . 6 |- (E.y(y = A /\ x e. y) <-> E.y(x e. y /\ y = A))
11	9, 10	bitr 151	. . . . 5 |- (x e. A <-> E.y(x e. y /\ y = A))
12		unpr.2	. . . . . . 7 |- B e. V
13	12	clel3 1375	. . . . . 6 |- (x e. B <-> E.y(y = B /\ x e. y))
14		exancom 736	. . . . . 6 |- (E.y(y = B /\ x e. y) <-> E.y(x e. y /\ y = B))
15	13, 14	bitr 151	. . . . 5 |- (x e. B <-> E.y(x e. y /\ y = B))
16	11, 15	orbi12i 216	. . . 4 |- ((x e. A \/ x e. B) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
17	1, 7, 16	3bitr4r 159	. . 3 |- ((x e. A \/ x e. B) <-> E.y(x e. y /\ y e. {A, B}))
18	17	biabi 1181	. 2 |- {x | (x e. A \/ x e. B)} = {x | E.y(x e. y /\ y e. {A, B})}
19		df-un 1490	. 2 |- (A u. B) = {x | (x e. A \/ x e. B)}
20		df-uni 1920	. 2 |- U.{A, B} = {x | E.y(x e. y /\ y e. {A, B})}
21	18, 19, 20	3eqtr4r 1127	1 |- U.{A, B} = (A u. B)

Syntax hints:   \/ wo 195   /\ wa 196  E.wex 678   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485  {cpr 1809  U.cuni 1919

This theorem is referenced by:  unop 1931  unisn 1932  unex 1949  dfchj3 5326

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-uni 1920

metamath.org