Theorem unipr 3912

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 1677	. . . 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 2806	. . . . . . . 8 |- y e. _V
3	2	elpr 3694	. . . . . . 7 |- ( y e. { A , B } <-> ( y = A \/ y = B ) )
4	3	anbi2i 457	. . . . . 6 |- ( ( x e. y /\ y e. { A , B } ) <-> ( x e. y /\ ( y = A \/ y = B ) ) )
5		andi 826	. . . . . 6 |- ( ( x e. y /\ ( y = A \/ y = B ) ) <-> ( ( x e. y /\ y = A ) \/ ( x e. y /\ y = B ) ) )
6	4, 5	bitri 184	. . . . 5 |- ( ( x e. y /\ y e. { A , B } ) <-> ( ( x e. y /\ y = A ) \/ ( x e. y /\ y = B ) ) )
7	6	exbii 1654	. . . 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 2942	. . . . . 6 |- ( x e. A <-> E. y ( y = A /\ x e. y ) )
10		exancom 1657	. . . . . 6 |- ( E. y ( y = A /\ x e. y ) <-> E. y ( x e. y /\ y = A ) )
11	9, 10	bitri 184	. . . . 5 |- ( x e. A <-> E. y ( x e. y /\ y = A ) )
12		unipr.2	. . . . . . 7 |- B e. _V
13	12	clel3 2942	. . . . . 6 |- ( x e. B <-> E. y ( y = B /\ x e. y ) )
14		exancom 1657	. . . . . 6 |- ( E. y ( y = B /\ x e. y ) <-> E. y ( x e. y /\ y = B ) )
15	13, 14	bitri 184	. . . . 5 |- ( x e. B <-> E. y ( x e. y /\ y = B ) )
16	11, 15	orbi12i 772	. . . 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 213	. . 3 |- ( ( x e. A \/ x e. B ) <-> E. y ( x e. y /\ y e. { A , B } ) )
18	17	abbii 2347	. 2 |- { x | ( x e. A \/ x e. B ) } = { x | E. y ( x e. y /\ y e. { A , B } ) }
19		df-un 3205	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
20		df-uni 3899	. 2 |- U. { A , B } = { x | E. y ( x e. y /\ y e. { A , B } ) }
21	18, 19, 20	3eqtr4ri 2263	1 |- U. { A , B } = ( A u. B )

Syntax hints:   /\ wa 104   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2202   {cab 2217   _Vcvv 2803   u. cun 3199   {cpr 3674   U.cuni 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-uni 3899

This theorem is referenced by:  uniprg  3913  unisn  3914  uniop  4354  unex  4544  bj-unex  16618