Theorem bdcuni 14488

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 14433	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 14431	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 14461	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2461	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1608	. . . . 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 2293	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 14455	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3810	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 14456	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1492   {cab 2163   E.wrex 2456   U.cuni 3809  BOUNDED wbdc 14452

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14425  ax-bdex 14431  ax-bdel 14433  ax-bdsb 14434

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-rex 2461  df-uni 3810  df-bdc 14453

This theorem is referenced by:  bj-uniex2  14528