Theorem bdcuni 13911

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 13856	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 13854	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 13884	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2454	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1601	. . . . 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 2286	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 13878	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3797	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 13879	1 |- BOUNDED U. x

Syntax hints:   /\ wa 103   E.wex 1485   {cab 2156   E.wrex 2449   U.cuni 3796  BOUNDED wbdc 13875

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bd0 13848  ax-bdex 13854  ax-bdel 13856  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rex 2454  df-uni 3797  df-bdc 13876

This theorem is referenced by:  bj-uniex2  13951