Theorem bdcuni 9996

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 9941	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 9939	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 9969	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2312	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1499	. . . . 5 |- ( E. z ( z e. x /\ y e. z ) <-> E. z ( y e. z /\ z e. x ) )
6	4, 5	bitri 173	. . . 4 |- ( E. z e. x y e. z <-> E. z ( y e. z /\ z e. x ) )
7	6	abbii 2153	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 9963	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3581	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 9964	1 |- BOUNDED U. x

Syntax hints:   /\ wa 97   E.wex 1381   {cab 2026   E.wrex 2307   U.cuni 3580  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdex 9939  ax-bdel 9941  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rex 2312  df-uni 3581  df-bdc 9961

This theorem is referenced by:  bj-uniex2  10036