Theorem unipr 3906

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)

Hypotheses

Ref	Expression
unipr.1	|- A e. _V
unipr.2	|- B e. _V

Assertion

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

Proof of Theorem unipr

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

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . . 8 |- y e. _V
2	1	elpr 3752	. . . . . . 7 |- (y e. {A, B} <-> (y = A \/ y = B))
3	2	anbi2i 675	. . . . . 6 |- ((x e. y /\ y e. {A, B}) <-> (x e. y /\ (y = A \/ y = B)))
4		andi 837	. . . . . 6 |- ((x e. y /\ (y = A \/ y = B)) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
5	3, 4	bitri 240	. . . . 5 |- ((x e. y /\ y e. {A, B}) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
6	5	exbii 1582	. . . 4 |- (E.y(x e. y /\ y e. {A, B}) <-> E.y((x e. y /\ y = A) \/ (x e. y /\ y = B)))
7		19.43 1605	. . . 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)))
8	6, 7	bitri 240	. . 3 |- (E.y(x e. y /\ y e. {A, B}) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
9		eluni 3895	. . 3 |- (x e. U.{A, B} <-> E.y(x e. y /\ y e. {A, B}))
10		elun 3221	. . . 4 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
11		unipr.1	. . . . . . 7 |- A e. _V
12	11	clel3 2978	. . . . . 6 |- (x e. A <-> E.y(y = A /\ x e. y))
13		exancom 1586	. . . . . 6 |- (E.y(y = A /\ x e. y) <-> E.y(x e. y /\ y = A))
14	12, 13	bitri 240	. . . . 5 |- (x e. A <-> E.y(x e. y /\ y = A))
15		unipr.2	. . . . . . 7 |- B e. _V
16	15	clel3 2978	. . . . . 6 |- (x e. B <-> E.y(y = B /\ x e. y))
17		exancom 1586	. . . . . 6 |- (E.y(y = B /\ x e. y) <-> E.y(x e. y /\ y = B))
18	16, 17	bitri 240	. . . . 5 |- (x e. B <-> E.y(x e. y /\ y = B))
19	14, 18	orbi12i 507	. . . 4 |- ((x e. A \/ x e. B) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
20	10, 19	bitri 240	. . 3 |- (x e. (A u. B) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
21	8, 9, 20	3bitr4i 268	. 2 |- (x e. U.{A, B} <-> x e. (A u. B))
22	21	eqriv 2350	1 |- U.{A, B} = (A u. B)

Syntax hints:   \/ wo 357   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   u. cun 3208  {cpr 3739  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893

This theorem is referenced by:  uniprg  3907  unisn  3908  uniintsn  3964