Theorem bdcpr 13753

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 13752	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 13752	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 13744	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3583	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 13726	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3114  {csn 3576  {cpr 3577  BOUNDED wbdc 13722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13695  ax-bdor 13698  ax-bdeq 13702  ax-bdsb 13704

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-un 3120  df-sn 3582  df-pr 3583  df-bdc 13723

This theorem is referenced by:  bdctp  13754  bdop  13757