Theorem bdcpr 14811

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 14810	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 14810	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 14802	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3601	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 14784	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3129  {csn 3594  {cpr 3595  BOUNDED wbdc 14780

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14753  ax-bdor 14756  ax-bdeq 14760  ax-bdsb 14762

This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-un 3135  df-sn 3600  df-pr 3601  df-bdc 14781

This theorem is referenced by:  bdctp  14812  bdop  14815