Theorem bdcuni 13063

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 13008	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 13006	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 13036	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2420	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1587	. . . . 5 |- ( E. z ( z e. x /\ y e. z ) <-> E. z ( y e. z /\ z e. x ) )
6	4, 5	bitri 183	. . . 4 |- ( E. z e. x y e. z <-> E. z ( y e. z /\ z e. x ) )
7	6	abbii 2253	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 13030	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3732	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 13031	1 |- BOUNDED U. x

Syntax hints:   /\ wa 103   E.wex 1468   {cab 2123   E.wrex 2415   U.cuni 3731  BOUNDED wbdc 13027

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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-bd0 13000  ax-bdex 13006  ax-bdel 13008  ax-bdsb 13009

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-rex 2420  df-uni 3732  df-bdc 13028

This theorem is referenced by:  bj-uniex2  13103