Theorem bdcpr 16690

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 16689	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 16689	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 16681	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3698	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 16663	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3211  {csn 3691  {cpr 3692  BOUNDED wbdc 16659

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216  ax-bd0 16632  ax-bdor 16635  ax-bdeq 16639  ax-bdsb 16641

This theorem depends on definitions:  df-bi 117  df-clab 2221  df-cleq 2227  df-clel 2230  df-un 3217  df-sn 3697  df-pr 3698  df-bdc 16660

This theorem is referenced by:  bdctp  16691  bdop  16694