Theorem bdcpr 15625

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 15624	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 15624	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 15616	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3630	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 15598	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3155  {csn 3623  {cpr 3624  BOUNDED wbdc 15594

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15567  ax-bdor 15570  ax-bdeq 15574  ax-bdsb 15576

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-un 3161  df-sn 3629  df-pr 3630  df-bdc 15595

This theorem is referenced by:  bdctp  15626  bdop  15629