Theorem bdcpr 15844

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 15843	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 15843	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 15835	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3640	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 15817	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 3164  {csn 3633  {cpr 3634  BOUNDED wbdc 15813

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187  ax-bd0 15786  ax-bdor 15789  ax-bdeq 15793  ax-bdsb 15795

This theorem depends on definitions:  df-bi 117  df-clab 2192  df-cleq 2198  df-clel 2201  df-un 3170  df-sn 3639  df-pr 3640  df-bdc 15814

This theorem is referenced by:  bdctp  15845  bdop  15848