Theorem uniprg 3811

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 3660	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	unieqd 3807	. . 3 |- ( x = A -> U. { x , y } = U. { A , y } )
3		uneq1 3274	. . 3 |- ( x = A -> ( x u. y ) = ( A u. y ) )
4	2, 3	eqeq12d 2185	. 2 |- ( x = A -> ( U. { x , y } = ( x u. y ) <-> U. { A , y } = ( A u. y ) ) )
5		preq2 3661	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	unieqd 3807	. . 3 |- ( y = B -> U. { A , y } = U. { A , B } )
7		uneq2 3275	. . 3 |- ( y = B -> ( A u. y ) = ( A u. B ) )
8	6, 7	eqeq12d 2185	. 2 |- ( y = B -> ( U. { A , y } = ( A u. y ) <-> U. { A , B } = ( A u. B ) ) )
9		vex 2733	. . 3 |- x e. _V
10		vex 2733	. . 3 |- y e. _V
11	9, 10	unipr 3810	. 2 |- U. { x , y } = ( x u. y )
12	4, 8, 11	vtocl2g 2794	1 |- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   u. cun 3119   {cpr 3584   U.cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  onun2  4474  unopn  12797