Theorem bdcun 16681

Description: The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdcdif.1	⊢ BOUNDED 𝐴
bdcdif.2	⊢ BOUNDED 𝐵

Assertion

Ref	Expression
bdcun	⊢ BOUNDED (𝐴 ∪ 𝐵)

Proof of Theorem bdcun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcdif.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 16665	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . 5 ⊢ BOUNDED 𝐵
4	3	bdeli 16665	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	2, 4	ax-bdor 16635	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)
6	5	bdcab 16668	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
7		df-un 3217	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
8	6, 7	bdceqir 16663	1 ⊢ BOUNDED (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 716   ∈ wcel 2205  {cab 2220   ∪ cun 3211  BOUNDED wbdc 16659

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 16632  ax-bdor 16635  ax-bdsb 16641

This theorem depends on definitions:  df-bi 117  df-clab 2221  df-cleq 2227  df-clel 2230  df-un 3217  df-bdc 16660

This theorem is referenced by:  bdcpr  16690  bdctp  16691  bdcsuc  16699