Theorem bdcuni 16816

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 16761	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 16759	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 16789	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2534	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1661	. . . . 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 2354	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 16783	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3931	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 16784	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1545   {cab 2224   E.wrex 2529   U.cuni 3930  BOUNDED wbdc 16780

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16753  ax-bdex 16759  ax-bdel 16761  ax-bdsb 16762

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rex 2534  df-uni 3931  df-bdc 16781

This theorem is referenced by: (None)