Theorem bdcuni 16646

Description: The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)

Assertion

Ref	Expression
bdcuni	|- BOUNDED U. x

Proof of Theorem bdcuni

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

Step	Hyp	Ref	Expression
1		ax-bdel 16591	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 16589	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 16619	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2526	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1657	. . . . 5 |- ( E. z ( z e. x /\ y e. z ) <-> E. z ( y e. z /\ z e. x ) )
6	4, 5	bitri 184	. . . 4 |- ( E. z e. x y e. z <-> E. z ( y e. z /\ z e. x ) )
7	6	abbii 2348	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 16613	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3915	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 16614	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1541   {cab 2218   E.wrex 2521   U.cuni 3914  BOUNDED wbdc 16610

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bd0 16583  ax-bdex 16589  ax-bdel 16591  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-rex 2526  df-uni 3915  df-bdc 16611

This theorem is referenced by:  bj-uniex2  16686