Theorem bdcpr 15006

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 15005	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 15005	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 14997	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3613	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 14979	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3141  {csn 3606  {cpr 3607  BOUNDED wbdc 14975

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544  ax-ext 2170  ax-bd0 14948  ax-bdor 14951  ax-bdeq 14955  ax-bdsb 14957

This theorem depends on definitions:  df-bi 117  df-clab 2175  df-cleq 2181  df-clel 2184  df-un 3147  df-sn 3612  df-pr 3613  df-bdc 14976

This theorem is referenced by:  bdctp  15007  bdop  15010