Theorem bdcpr 10662

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 10661	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 10661	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 10653	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3405	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 10635	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 2971  {csn 3398  {cpr 3399  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604  ax-bdor 10607  ax-bdeq 10611  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-un 2977  df-sn 3404  df-pr 3405  df-bdc 10632

This theorem is referenced by:  bdctp  10663  bdop  10666