Theorem bdcpr 16234

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 16233	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 16233	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 16225	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3673	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 16207	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3195  {csn 3666  {cpr 3667  BOUNDED wbdc 16203

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16176  ax-bdor 16179  ax-bdeq 16183  ax-bdsb 16185

This theorem depends on definitions:  df-bi 117  df-clab 2216  df-cleq 2222  df-clel 2225  df-un 3201  df-sn 3672  df-pr 3673  df-bdc 16204

This theorem is referenced by:  bdctp  16235  bdop  16238