Theorem bdcpr 9991

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 9990	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 9990	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 9982	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3382	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 9964	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 2915  {csn 3375  {cpr 3376  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdor 9936  ax-bdeq 9940  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-un 2922  df-sn 3381  df-pr 3382  df-bdc 9961

This theorem is referenced by:  bdctp  9992  bdop  9995