Theorem bdcuni 16471

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 16416	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 16414	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 16444	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2516	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1656	. . . . 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 2347	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 16438	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3894	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 16439	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1540   {cab 2217   E.wrex 2511   U.cuni 3893  BOUNDED wbdc 16435

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bd0 16408  ax-bdex 16414  ax-bdel 16416  ax-bdsb 16417

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-rex 2516  df-uni 3894  df-bdc 16436

This theorem is referenced by:  bj-uniex2  16511