Theorem bdcuni 15816

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 15761	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 15759	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 15789	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2490	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1631	. . . . 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 2321	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 15783	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3851	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 15784	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1515   {cab 2191   E.wrex 2485   U.cuni 3850  BOUNDED wbdc 15780

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bd0 15753  ax-bdex 15759  ax-bdel 15761  ax-bdsb 15762

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-rex 2490  df-uni 3851  df-bdc 15781

This theorem is referenced by:  bj-uniex2  15856