Theorem bdcuni 10667

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 10612	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 10610	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 10640	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2354	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1539	. . . . 5 |- ( E. z ( z e. x /\ y e. z ) <-> E. z ( y e. z /\ z e. x ) )
6	4, 5	bitri 182	. . . 4 |- ( E. z e. x y e. z <-> E. z ( y e. z /\ z e. x ) )
7	6	abbii 2194	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 10634	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3602	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 10635	1 |- BOUNDED U. x

Syntax hints:   /\ wa 102   E.wex 1421   {cab 2067   E.wrex 2349   U.cuni 3601  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdex 10610  ax-bdel 10612  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-rex 2354  df-uni 3602  df-bdc 10632

This theorem is referenced by:  bj-uniex2  10707