Theorem bdctp 15528

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 15527	. . 3 ⊢ BOUNDED {𝑥, 𝑦}
2		bdcsn 15526	. . 3 ⊢ BOUNDED {𝑧}
3	1, 2	bdcun 15518	. 2 ⊢ BOUNDED ({𝑥, 𝑦} ∪ {𝑧})
4		df-tp 3631	. 2 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥, 𝑦} ∪ {𝑧})
5	3, 4	bdceqir 15500	1 ⊢ BOUNDED {𝑥, 𝑦, 𝑧}

Syntax hints:   ∪ cun 3155  {csn 3623  {cpr 3624  {ctp 3625  BOUNDED wbdc 15496

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15469  ax-bdor 15472  ax-bdeq 15476  ax-bdsb 15478

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-un 3161  df-sn 3629  df-pr 3630  df-tp 3631  df-bdc 15497

This theorem is referenced by: (None)