Theorem uniprg 3865

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 3710	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	unieqd 3861	. . 3 |- ( x = A -> U. { x , y } = U. { A , y } )
3		uneq1 3320	. . 3 |- ( x = A -> ( x u. y ) = ( A u. y ) )
4	2, 3	eqeq12d 2220	. 2 |- ( x = A -> ( U. { x , y } = ( x u. y ) <-> U. { A , y } = ( A u. y ) ) )
5		preq2 3711	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	unieqd 3861	. . 3 |- ( y = B -> U. { A , y } = U. { A , B } )
7		uneq2 3321	. . 3 |- ( y = B -> ( A u. y ) = ( A u. B ) )
8	6, 7	eqeq12d 2220	. 2 |- ( y = B -> ( U. { A , y } = ( A u. y ) <-> U. { A , B } = ( A u. B ) ) )
9		vex 2775	. . 3 |- x e. _V
10		vex 2775	. . 3 |- y e. _V
11	9, 10	unipr 3864	. 2 |- U. { x , y } = ( x u. y )
12	4, 8, 11	vtocl2g 2837	1 |- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   u. cun 3164   {cpr 3634   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851

This theorem is referenced by:  onun2  4539  unopn  14510