Theorem bdcpr 16811

Description: The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)

Assertion

Ref	Expression
bdcpr	|- BOUNDED { x , y }

Proof of Theorem bdcpr

Step	Hyp	Ref	Expression
1		bdcsn 16810	. . 3 |- BOUNDED { x }
2		bdcsn 16810	. . 3 |- BOUNDED { y }
3	1, 2	bdcun 16802	. 2 |- BOUNDED ( { x } u. { y } )
4		df-pr 3712	. 2 |- { x , y } = ( { x } u. { y } )
5	3, 4	bdceqir 16784	1 |- BOUNDED { x , y }

Syntax hints:   u. cun 3218   {csn 3705   {cpr 3706  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-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16753  ax-bdor 16756  ax-bdeq 16760  ax-bdsb 16762

This theorem depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-un 3224  df-sn 3711  df-pr 3712  df-bdc 16781

This theorem is referenced by:  bdctp  16812  bdop  16815