Theorem unipr 4176

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		19.43 1701	. . . 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		vex 3037	. . . . . . . 8 |- y e. _V
3	2	elpr 3962	. . . . . . 7 |- ( y e. { A , B } <-> ( y = A \/ y = B ) )
4	3	anbi2i 692	. . . . . 6 |- ( ( x e. y /\ y e. { A , B } ) <-> ( x e. y /\ ( y = A \/ y = B ) ) )
5		andi 865	. . . . . 6 |- ( ( x e. y /\ ( y = A \/ y = B ) ) <-> ( ( x e. y /\ y = A ) \/ ( x e. y /\ y = B ) ) )
6	4, 5	bitri 249	. . . . 5 |- ( ( x e. y /\ y e. { A , B } ) <-> ( ( x e. y /\ y = A ) \/ ( x e. y /\ y = B ) ) )
7	6	exbii 1675	. . . 4 |- ( E. y ( x e. y /\ y e. { A , B } ) <-> E. y ( ( x e. y /\ y = A ) \/ ( x e. y /\ y = B ) ) )
8		unipr.1	. . . . . . 7 |- A e. _V
9	8	clel3 3163	. . . . . 6 |- ( x e. A <-> E. y ( y = A /\ x e. y ) )
10		exancom 1679	. . . . . 6 |- ( E. y ( y = A /\ x e. y ) <-> E. y ( x e. y /\ y = A ) )
11	9, 10	bitri 249	. . . . 5 |- ( x e. A <-> E. y ( x e. y /\ y = A ) )
12		unipr.2	. . . . . . 7 |- B e. _V
13	12	clel3 3163	. . . . . 6 |- ( x e. B <-> E. y ( y = B /\ x e. y ) )
14		exancom 1679	. . . . . 6 |- ( E. y ( y = B /\ x e. y ) <-> E. y ( x e. y /\ y = B ) )
15	13, 14	bitri 249	. . . . 5 |- ( x e. B <-> E. y ( x e. y /\ y = B ) )
16	11, 15	orbi12i 519	. . . 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	3bitr4ri 278	. . 3 |- ( ( x e. A \/ x e. B ) <-> E. y ( x e. y /\ y e. { A , B } ) )
18	17	abbii 2516	. 2 |- { x | ( x e. A \/ x e. B ) } = { x | E. y ( x e. y /\ y e. { A , B } ) }
19		df-un 3394	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
20		df-uni 4164	. 2 |- U. { A , B } = { x | E. y ( x e. y /\ y e. { A , B } ) }
21	18, 19, 20	3eqtr4ri 2422	1 |- U. { A , B } = ( A u. B )

Syntax hints:   \/ wo 366   /\ wa 367   = wceq 1399   E.wex 1620   e. wcel 1826   {cab 2367   _Vcvv 3034   u. cun 3387   {cpr 3946   U.cuni 4163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-un 3394  df-sn 3945  df-pr 3947  df-uni 4164

This theorem is referenced by:  uniprg  4177  unisn  4178  uniintsn  4237  uniop  4664  unex  6497  rankxplim  8210  mrcun  15029  indistps  19597  indistps2  19598  leordtval2  19799  ex-uni  25268  fouriersw  32180