Theorem bdctp 16768

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 16767	. . 3 ⊢ BOUNDED {𝑥, 𝑦}
2		bdcsn 16766	. . 3 ⊢ BOUNDED {𝑧}
3	1, 2	bdcun 16758	. 2 ⊢ BOUNDED ({𝑥, 𝑦} ∪ {𝑧})
4		df-tp 3702	. 2 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥, 𝑦} ∪ {𝑧})
5	3, 4	bdceqir 16740	1 ⊢ BOUNDED {𝑥, 𝑦, 𝑧}

Syntax hints:   ∪ cun 3212  {csn 3694  {cpr 3695  {ctp 3696  BOUNDED wbdc 16736

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216  ax-bd0 16709  ax-bdor 16712  ax-bdeq 16716  ax-bdsb 16718

This theorem depends on definitions:  df-bi 117  df-clab 2221  df-cleq 2227  df-clel 2230  df-un 3218  df-sn 3700  df-pr 3701  df-tp 3702  df-bdc 16737

This theorem is referenced by: (None)