Theorem bdcpr 16641

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 16640	. . 3 |- BOUNDED { x }
2		bdcsn 16640	. . 3 |- BOUNDED { y }
3	1, 2	bdcun 16632	. 2 |- BOUNDED ( { x } u. { y } )
4		df-pr 3696	. 2 |- { x , y } = ( { x } u. { y } )
5	3, 4	bdceqir 16614	1 |- BOUNDED { x , y }

Syntax hints:   u. cun 3209   {csn 3689   {cpr 3690  BOUNDED wbdc 16610

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214  ax-bd0 16583  ax-bdor 16586  ax-bdeq 16590  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-un 3215  df-sn 3695  df-pr 3696  df-bdc 16611

This theorem is referenced by:  bdctp  16642  bdop  16645