Theorem bdcpr 11408

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 11407	. . 3 ⊢ BOUNDED {𝑥}
2		bdcsn 11407	. . 3 ⊢ BOUNDED {𝑦}
3	1, 2	bdcun 11399	. 2 ⊢ BOUNDED ({𝑥} ∪ {𝑦})
4		df-pr 3448	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
5	3, 4	bdceqir 11381	1 ⊢ BOUNDED {𝑥, 𝑦}

Syntax hints:   ∪ cun 2995  {csn 3441  {cpr 3442  BOUNDED wbdc 11377

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11350  ax-bdor 11353  ax-bdeq 11357  ax-bdsb 11359

This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-un 3001  df-sn 3447  df-pr 3448  df-bdc 11378

This theorem is referenced by:  bdctp  11409  bdop  11412