Theorem bdcuni 15428

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 15373	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 15371	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 15401	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2478	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1619	. . . . 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 2309	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 15395	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3837	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 15396	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1503   {cab 2179   E.wrex 2473   U.cuni 3836  BOUNDED wbdc 15392

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15365  ax-bdex 15371  ax-bdel 15373  ax-bdsb 15374

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-rex 2478  df-uni 3837  df-bdc 15393

This theorem is referenced by:  bj-uniex2  15468