Theorem bdcuni 15522

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 15467	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdex 15465	. . . 4 |- BOUNDED E. z e. x y e. z
3	2	bdcab 15495	. . 3 |- BOUNDED { y | E. z e. x y e. z }
4		df-rex 2481	. . . . 5 |- ( E. z e. x y e. z <-> E. z ( z e. x /\ y e. z ) )
5		exancom 1622	. . . . 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 2312	. . 3 |- { y | E. z e. x y e. z } = { y | E. z ( y e. z /\ z e. x ) }
8	3, 7	bdceqi 15489	. 2 |- BOUNDED { y | E. z ( y e. z /\ z e. x ) }
9		df-uni 3840	. 2 |- U. x = { y | E. z ( y e. z /\ z e. x ) }
10	8, 9	bdceqir 15490	1 |- BOUNDED U. x

Syntax hints:   /\ wa 104   E.wex 1506   {cab 2182   E.wrex 2476   U.cuni 3839  BOUNDED wbdc 15486

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15459  ax-bdex 15465  ax-bdel 15467  ax-bdsb 15468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rex 2481  df-uni 3840  df-bdc 15487

This theorem is referenced by:  bj-uniex2  15562