Theorem bdcpr 16487

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 16486	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 16486	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 16478	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3676	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 16460	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3198  {csn 3669  {cpr 3670  BOUNDED wbdc 16456

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213  ax-bd0 16429  ax-bdor 16432  ax-bdeq 16436  ax-bdsb 16438

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-cleq 2224  df-clel 2227  df-un 3204  df-sn 3675  df-pr 3676  df-bdc 16457

This theorem is referenced by:  bdctp  16488  bdop  16491