Theorem bdcpr 17063

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 17062	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 17062	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 17054	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3716	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 17036	1 ⊢ BOUNDED {𝑥, 𝑦}

This proof depends on syntax axioms:   ∪ cun 3218  {csn 3709  {cpr 3710  BOUNDED wbdc 17032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 17005  ax-bdor 17008  ax-bdeq 17012  ax-bdsb 17014

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-un 3224  df-sn 3715  df-pr 3716  df-bdc 17033

This theorem is used by:  bdctp  17064  bdop  17067