Theorem bdcpr 13058

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 13057	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 13057	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 13049	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3529	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 13031	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3064  {csn 3522  {cpr 3523  BOUNDED wbdc 13027

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-bd0 13000  ax-bdor 13003  ax-bdeq 13007  ax-bdsb 13009

This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-un 3070  df-sn 3528  df-pr 3529  df-bdc 13028

This theorem is referenced by:  bdctp  13059  bdop  13062