Theorem bdcpr 13906

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 13905	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 13905	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 13897	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3590	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 13879	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3119  {csn 3583  {cpr 3584  BOUNDED wbdc 13875

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-bd0 13848  ax-bdor 13851  ax-bdeq 13855  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-clab 2157  df-cleq 2163  df-clel 2166  df-un 3125  df-sn 3589  df-pr 3590  df-bdc 13876

This theorem is referenced by:  bdctp  13907  bdop  13910