Metamath Proof Explorer

Theorem uniprgOLD

Description: Obsolete version of unipr as of 1-Sep-2024. (Contributed by NM, 25-Aug-2006) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

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