Theorem uniprg 3826

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)

Assertion

Ref	Expression
uniprg	|- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) )

Proof of Theorem uniprg

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

Step	Hyp	Ref	Expression
1		preq1 3671	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	unieqd 3822	. . 3 |- ( x = A -> U. { x , y } = U. { A , y } )
3		uneq1 3284	. . 3 |- ( x = A -> ( x u. y ) = ( A u. y ) )
4	2, 3	eqeq12d 2192	. 2 |- ( x = A -> ( U. { x , y } = ( x u. y ) <-> U. { A , y } = ( A u. y ) ) )
5		preq2 3672	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	unieqd 3822	. . 3 |- ( y = B -> U. { A , y } = U. { A , B } )
7		uneq2 3285	. . 3 |- ( y = B -> ( A u. y ) = ( A u. B ) )
8	6, 7	eqeq12d 2192	. 2 |- ( y = B -> ( U. { A , y } = ( A u. y ) <-> U. { A , B } = ( A u. B ) ) )
9		vex 2742	. . 3 |- x e. _V
10		vex 2742	. . 3 |- y e. _V
11	9, 10	unipr 3825	. 2 |- U. { x , y } = ( x u. y )
12	4, 8, 11	vtocl2g 2803	1 |- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   u. cun 3129   {cpr 3595   U.cuni 3811

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-uni 3812

This theorem is referenced by:  onun2  4491  unopn  13590