Theorem bdcpr 16006

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 16005	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 16005	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 15997	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3650	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 15979	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3172  {csn 3643  {cpr 3644  BOUNDED wbdc 15975

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-bd0 15948  ax-bdor 15951  ax-bdeq 15955  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-clab 2194  df-cleq 2200  df-clel 2203  df-un 3178  df-sn 3649  df-pr 3650  df-bdc 15976

This theorem is referenced by:  bdctp  16007  bdop  16010