Theorem bdcuni 16011

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 15956	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 15954	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 15984	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2492	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1632	. . . . 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 2323	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 15978	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3865	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 15979	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1516   {cab 2193   E.wrex 2487   U.cuni 3864  BOUNDED wbdc 15975

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bd0 15948  ax-bdex 15954  ax-bdel 15956  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-rex 2492  df-uni 3865  df-bdc 15976

This theorem is referenced by:  bj-uniex2  16051