Theorem bdcuni 16239

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 16184	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 16182	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 16212	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2514	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1654	. . . . 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 2345	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 16206	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3889	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 16207	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1538   {cab 2215   E.wrex 2509   U.cuni 3888  BOUNDED wbdc 16203

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16176  ax-bdex 16182  ax-bdel 16184  ax-bdsb 16185

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-rex 2514  df-uni 3889  df-bdc 16204

This theorem is referenced by:  bj-uniex2  16279