Theorem bdctp 15102

Description: The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)

Assertion

Ref	Expression
bdctp	⊢ BOUNDED {𝑥, 𝑦, 𝑧}

Proof of Theorem bdctp

Step	Hyp	Ref	Expression
1		bdcpr 15101	. . 3 ⊢ BOUNDED {𝑥, 𝑦}
2		bdcsn 15100	. . 3 ⊢ BOUNDED {𝑧}
3	1, 2	bdcun 15092	. 2 ⊢ BOUNDED ({𝑥, 𝑦} ∪ {𝑧})
4		df-tp 3615	. 2 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥, 𝑦} ∪ {𝑧})
5	3, 4	bdceqir 15074	1 ⊢ BOUNDED {𝑥, 𝑦, 𝑧}

Syntax hints:   ∪ cun 3142  {csn 3607  {cpr 3608  {ctp 3609  BOUNDED wbdc 15070

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 15043  ax-bdor 15046  ax-bdeq 15050  ax-bdsb 15052

This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-un 3148  df-sn 3613  df-pr 3614  df-tp 3615  df-bdc 15071

This theorem is referenced by: (None)