Theorem bdcpr 10913

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 10912	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 10912	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 10904	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3424	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 10886	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 2981  {csn 3417  {cpr 3418  BOUNDED wbdc 10882

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065  ax-bd0 10855  ax-bdor 10858  ax-bdeq 10862  ax-bdsb 10864

This theorem depends on definitions:  df-bi 115  df-clab 2070  df-cleq 2076  df-clel 2079  df-un 2987  df-sn 3423  df-pr 3424  df-bdc 10883

This theorem is referenced by:  bdctp  10914  bdop  10917