Theorem bdcpr 15081

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 15080	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 15080	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 15072	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3614	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 15054	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3142  {csn 3607  {cpr 3608  BOUNDED wbdc 15050

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 15023  ax-bdor 15026  ax-bdeq 15030  ax-bdsb 15032

This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-un 3148  df-sn 3613  df-pr 3614  df-bdc 15051

This theorem is referenced by:  bdctp  15082  bdop  15085