Theorem bdcuni 13758

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 13703	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 13701	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 13731	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2450	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1596	. . . . 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 2282	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 13725	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3790	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 13726	1 |- BOUNDED U. x

Syntax hints:   /\ wa 103   E.wex 1480   {cab 2151   E.wrex 2445   U.cuni 3789  BOUNDED wbdc 13722

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bd0 13695  ax-bdex 13701  ax-bdel 13703  ax-bdsb 13704

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-rex 2450  df-uni 3790  df-bdc 13723

This theorem is referenced by:  bj-uniex2  13798