Theorem bdcuni 13245

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 13190	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 13188	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 13218	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2423	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1588	. . . . 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 2256	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 13212	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3745	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 13213	1 |- BOUNDED U. x

Syntax hints:   /\ wa 103   E.wex 1469   {cab 2126   E.wrex 2418   U.cuni 3744  BOUNDED wbdc 13209

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bd0 13182  ax-bdex 13188  ax-bdel 13190  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-rex 2423  df-uni 3745  df-bdc 13210

This theorem is referenced by:  bj-uniex2  13285